Everyday examples of cohomology

Everyday examples of cohomology

Here’s a grab-bag of “cohomology in the wild”—ways the idea shows up in everyday-ish things. Think: local rules that don’t quite fit together globally, and the little “residue” of that mismatch is a cohomology class.

• Setting one clock for a whole network (H¹ of a graph). Suppose edges of a network store measured time differences between neighboring devices. If those differences come from a single global clock, their sums around every cycle are 0. If some loop sums to +3 seconds, you can’t assign one consistent clock—your loop sum is a nonzero cohomology class.

• Kirchhoff’s laws in circuits (graph cohomology). Node voltages are 0-cochains; edge voltage drops are 1-cochains. Taking differences (“gradient”) sends node voltages → edge drops. A pattern of drops is exact iff every loop sums to 0 (KVL). Batteries/induced EMFs make nonzero loop sums: that’s a nontrivial class in $H^1$ measuring how much “potential” fails to exist globally.

• Wind around a chimney / water around a drain (de Rham $H^1$). Away from the chimney the field can be curl-free (closed), yet the circulation $\oint$ around the chimney isn’t 0. There’s no global potential function because the domain has a hole. That “circulation around the hole” lives in $H^1$ and counts independent loops.

• Magnetic/electric induction loops. Faraday’s law $\oint E\cdot dl=-\frac{d{\mathrm{\Phi }}_B}{dt}$ says the electric field around a changing magnetic flux isn’t a pure gradient globally—even if it looks locally conservative. The cohomology class detects the “global twist” forced by the changing flux.

• Panorama/map stitching (Čech $H^1$). You align overlapping photos (local matches). If the pairwise alignments are perfectly compatible, they come from one global placement. If you go around a cycle of overlaps and come back rotated/shifted, that loop defect is a 1-cocycle that isn’t a coboundary—nonzero class says “no globally consistent stitching without a cut.”

• Reconstructing an image from its gradients (Poisson editing). Given desired pixel-gradients, you can recover an image iff those gradients are an exact field. If loop integrals don’t vanish, you get seams unless you cut the domain (killing $H^1$) or adjust gradients to kill the cohomology.

• GPS path independence & “conservative costs.” If road “efforts” (fuel/time differentials) are integrable to a single potential, the cost from A→B is path-independent. One-way streets, toll loops, or wind currents can create loop imbalances: those are precisely the obstructions measured in $H^1$.

• “Counting connected pieces” ( $H^0$ ). $H^0$ is the simplest: it counts connected components. In practice: how many islands are there in your maze/map/graph? That’s the dimension of $H^0$.

• Global balance laws (top cohomology). Integrating a density over a region gives a number that only depends on the region’s “top” cohomology class (think total mass/charge). Change coordinates, wiggle things—integrals of exact top-forms vanish; only the cohomology class survives.

One-line motto: cohomology measures whether “locally defined stuff” (differences, gradients, patches, phases) can be glued into a single global object—and if not, how it fails.

If you want, we can pick one (say, circuits or the panorama example) and work through a tiny concrete calculation.

Ancient Ocean Sea Concept

The ancient idea of the "Ocean sea"

The ancient idea of the "Ocean Sea" (often called Mare Oceanum in Latin or Ὠκεανός in Greek) was the belief—common in many classical and medieval cultures—that all the known landmasses were surrounded by one vast, continuous body of water.

Here’s how it played out historically:

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1. Origins in Ancient Greece

• Homer’s world (8th century BCE): In the Iliad and Odyssey, the “Ocean” (Okeanos) is both a god and a river-like sea encircling the Earth. It wasn’t thought of as just salt water, but as a primal, life-giving boundary beyond which lay the unknown.
• Geographers like Hecataeus and Herodotus: They preserved the idea that the “inhabited world” (oikoumene) was like an island surrounded by this encircling ocean.
• Eratosthenes and Strabo: More scientific approaches emerged, but the encircling ocean still appeared on maps—sometimes in circular form.

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2. Roman and Medieval Continuation

• The Romans kept the concept, calling it Oceanus, distinguishing it from inner seas (like the Mediterranean).
• Medieval European mappae mundi often show the continents (Asia, Europe, Africa) ringed by the “Ocean Sea,” with monsters, islands, and strange marvels beyond.
• Norse cosmology had a similar idea—miðgarðr (the human world) surrounded by a great sea inhabited by the Midgard Serpent.

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3. The “Ocean Sea” in the Age of Discovery

• By the 15th century, Europeans used “Ocean Sea” (Mar Oceana in Spanish, Mar Océano in Portuguese) to mean the Atlantic Ocean in particular.
• Columbus famously set out across the “Ocean Sea” in 1492, thinking it was a relatively narrow waterway between Europe and Asia.
• This reflected a lingering medieval mindset: the ocean was vast but finite, a ring to be crossed to reach the other side of the world.

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4. Symbolism and Cultural Meaning

• The “Ocean Sea” was both a geographical boundary and a psychological one—the edge of the known world.
• It symbolized mystery, danger, and the limits of human exploration.
• In maps and literature, it was often populated with sea serpents, islands of the blessed, or portals to other realms.

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If you like, I can sketch you a simple diagram showing how ancient people visualized the “Ocean Sea” as a ring around the continents—it makes the concept much clearer visually.

Make an ALU using NAND gates

How do you make an ALU using nand gates?

Fun project! The clean way to “make an ALU from NAND gates” is to (1) build the basic gates from NANDs, (2) build slightly bigger blocks (XOR, full adder, multiplexers) from those, then (3) wire a 1‑bit ALU “slice” and replicate it N times.

Here’s a compact blueprint you can follow.

1) Everything from NAND

NAND is functionally complete, so first express the usual gates:

• NOT: ¬A = NAND(A, A)
• AND: A ∧ B = ¬NAND(A, B) = NAND(NAND(A, B), NAND(A, B))
• OR (via De Morgan): A ∨ B = NAND(¬A, ¬B) = NAND(NAND(A, A), NAND(B, B))
• XOR (minimal 4‑NAND version): t1 = NAND(A, B) t2 = NAND(A, t1) t3 = NAND(B, t1) A ⊕ B = NAND(t2, t3)
• 2:1 MUX, y = s̄·d0 + s·d1, in NANDs: s̄ = NAND(s, s) u = NAND(d0, s̄) v = NAND(d1, s) y = NAND(u, v)

These five are the only “primitives” you’ll need.

2) Full adder from NANDs

One common approach is “XOR‑XOR‑AND/OR”:

• p = A ⊕ B (4 NANDs as above)
• sum = p ⊕ Cin (another 4 NANDs)
• c1 = A ∧ B (2 NANDs for AND)
• c2 = A ∧ Cin (2 NANDs)
• c3 = B ∧ Cin (2 NANDs)
• Cout = c1 ∨ c2 ∨ c3 (OR = NAND(NOT,NOT); chain two ORs with NANDs)

Gate count isn’t critical unless you’re optimizing, but this maps cleanly and is easy to debug. (There’s also a well‑known 9‑NAND full‑adder topology if you’re chasing minimum count.)

3) One 1‑bit ALU slice

Give the slice two operands (Ai, Bi), a carry‑in (Ci), and control bits op[2:0]. Compute the candidate results in parallel, then use a MUX to pick one:

Common ops (examples; choose what you need):

• AND: Ai ∧ Bi
• OR: Ai ∨ Bi
• XOR: Ai ⊕ Bi
• ADD: Si = Ai ⊕ Bi ⊕ Ci, Cout from the full adder
• SUB: do two’s complement: add Ai + (¬Bi) + 1 Implementation trick: feed Bi′ = (bInvert ? ¬Bi : Bi) into the adder and set Ci = bInvert for the LSB. This is exactly how the MIPS/LC‑3 textbook ALUs do it.
• PASSA / PASSB: route through a MUX
• SLT (set‑less‑than): in the MSB slice, take the sign of A − B and route it into bit0 via a dedicated path (others output 0). You only need this if you want comparisons.

Inside the slice:

1. Make Bi′ = MUX(bInvert, Bi, ¬Bi) using the NAND MUX above.

2. Build the full adder with (Ai, Bi′, Ci) → (Sum, Co).

3. Compute logic results (AND/OR/XOR) with the NAND recipes.

4. Select the output with a 4:1 or 8:1 MUX in NANDs, controlled by op[2:0].

   • For example, use two cascaded 2:1 NAND‑MUXes to build a 4:1.

Example encoding (use any you like):

• 000 → AND
• 001 → OR
• 010 → XOR
• 011 → ADD
• 100 → SUB (i.e., same data path as ADD, but assert bInvert=1; for LSB, Ci=1)
• 101 → PASS A
• 110 → PASS B
• 111 → SLT (only meaningful in bit0; others drive 0)

Outputs per slice: result bit (Ri) and carry‑out (Ci+1). Chain Ci+1 → next slice’s Ci to form an N‑bit ripple‑carry ALU.

4) Scale to N bits

Replicate the slice N times (4, 8, 16, 32…):

• Tie all op[2:0] and bInvert lines in parallel across slices.

• For ADD/SUB:

  • LSB: Ci = bInvert (0 for ADD, 1 for SUB)
  • Middle bits: Ci from previous slice
  • MSB: exposes Cout and (optionally) the signed overflow flag: V = Cout ⊕ Cn−1

• For SLT: compute A − B in the background (SUB mode), capture the MSB’s sign (or overflow‑corrected less‑than), and route it to bit0 through the 111 MUX selection. Others drive 0 for 111.

5) Practical notes

• Fan‑in: stick to 2‑input NANDs. If you need wider ORs, tree them (pairwise with NAND‑based OR).
• Timing: this is a ripple‑carry design; it’s simple but slowish. If you care about speed, add carry‑lookahead later (still doable in NAND‑only logic, just more gates).
• Reset/flags: you can add Zero flag by NOR‑reducing all result bits (NOR = NOT(OR) using NANDs), Sign from MSB, Carry from last Cout, Overflow as above.

6) Minimal NAND‑only Verilog (illustrative)

If you want to simulate before wiring:

// 1-bit slice, NAND-only style (illustrative subset: AND/OR/XOR/ADD)
module alu1 (
  input  a, b, cin,
  input  op0, op1, op2, // select lines
  input  bInvert,       // 0:add/use b, 1:sub/use ~b
  output result,
  output cout
);
  wire nb, bsel, t1,t2,t3, xor_ab, p1,p2,p3, c1,c2,c3, sum;

  // NOT b
  nand(nb, b, b);
  // bsel = MUX(bInvert ? ~b : b)
  wire ninv; nand(ninv, bInvert, bInvert);
  wire m0, m1;
  nand(m0, nb, ninv);     // ~b & ~bInvert
  nand(m1, b,  bInvert);  //  b &  bInvert
  nand(bsel, m0, m1);     // OR via NAND-NAND → MUX output

  // XOR a ^ bsel (4-NAND)
  nand(t1, a, bsel);
  nand(t2, a, t1);
  nand(t3, bsel, t1);
  nand(xor_ab, t2, t3);

  // SUM = xor_ab ^ cin (4-NAND more)
  wire s1,s2,s3;
  nand(s1, xor_ab, cin);
  nand(s2, xor_ab, s1);
  nand(s3, cin,    s1);
  nand(sum, s2, s3);

  // Carry = (a&bsel) | (a&cin) | (bsel&cin)
  // ANDs in NAND form
  wire ab_n, ac_n, bc_n, ab, ac, bc, c12_n, c12, c_all;
  nand(ab_n, a, bsel); nand(ab, ab_n, ab_n);
  nand(ac_n, a, cin);  nand(ac, ac_n, ac_n);
  nand(bc_n, bsel, cin); nand(bc, bc_n, bc_n);
  // OR three terms: (ab | ac) | bc
  wire abac_n;                  // OR via NAND of NOTs
  // NOT ab / NOT ac
  wire nab, nac; nand(nab, ab, ab); nand(nac, ac, ac);
  nand(abac_n, nab, nac);       // ~(~ab & ~ac) = ab|ac
  wire abac; nand(abac, abac_n, abac_n);
  // (ab|ac)|bc
  wire nabac, nbc; nand(nabac, abac, abac); nand(nbc, bc, bc);
  wire carr_or_n; nand(carr_or_n, nabac, nbc);
  wire carry; nand(carry, carr_or_n, carr_or_n);

  assign cout = carry;

  // Logic ops:
  wire and_n, or_n, a_and_b, a_or_b, a_xor_b;
  // AND
  nand(and_n, a, b); nand(a_and_b, and_n, and_n);
  // OR
  wire na, nb2; nand(na, a, a); nand(nb2, b, b);
  nand(or_n, na, nb2); nand(a_or_b, or_n, or_n);
  // XOR (reuse form with original a,b)
  wire u1,u2,u3; nand(u1, a, b); nand(u2, a, u1); nand(u3, b, u1); nand(a_xor_b, u2, u3);

  // 4:1 MUX (use two 2:1 MUX stages) to choose among AND/OR/XOR/SUM
  // Here: op1:op0 select (ignore op2 in this minimal example)
  function automatic mux2 (input d0, input d1, input s);
    // synthesizable macro in real toolchains; shown behaviorally here
    mux2 = s ? d1 : d0;
  endfunction

  wire mA = mux2(a_and_b, a_or_b, op0);
  wire mB = mux2(a_xor_b, sum,    op0);
  assign result = mux2(mA, mB, op1);
endmodule

(When you build it purely from nand primitives, replace the behavioral mux2 with the 2:1 NAND‑MUX shown earlier.)

7) How to wire a 4‑bit ALU quickly

• Instantiate 4 slices: bit0..bit3.
• For ADD: set bInvert=0, bit0.cin=0. For SUB: set bInvert=1, bit0.cin=1.
• Chain carries: bit0.cout→bit1.cin→bit2.cin→bit3.cin.
• Use the same op[2:0] for all slices.
• Optional flags: Z = NOR of all result bits; C = bit3.cout; V = c3_out ⊕ c2_out; N = result[3].

If you want, tell me what bit‑width and exact ops you need and I’ll map the control table and draw you a gate‑level slice with the exact NAND count.

Rings vs semirings

Rings vs semirings

Overview A semiring is an algebraic structure with two binary operations (usually called “addition” and “multiplication”) satisfying most of the familiar ring axioms except that additive inverses need not exist. A ring, by contrast, does require that every element have an additive inverse. Below is a side-by-side comparison of their defining properties, followed by examples and some remarks on when each arises in mathematics and computer science.

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1. Definitions and Axioms

Property	Semiring	Ring
Underlying set	$S$	$R$
Addition $(+,0)$	Commutative monoid:$\mathrm{\forall }a,b,c\in S$,$a+(b+c)=(a+b)+c$,$a+b=b+a$,$0+a=a$.	Abelian group:All semiring properties plus$\mathrm{\forall }a\mathrm{\exists }(-a)$ such that $a+(-a)=0$.
Multiplication $(\cdot ,1)$	Monoid:$\mathrm{\forall }a,b,c\in S$,$a\cdot (b\cdot c)=(a\cdot b)\cdot c$,often with identity $1$.	Monoid:Same as semiring. Often required to have $1$.
Distributivity	$\mathrm{\forall }a,b,c$,$a\cdot (b+c)=a\cdot b+a\cdot c$,$(b+c)\cdot a=b\cdot a+c\cdot a$.	Same as semiring.
Zero-absorption	$\mathrm{\forall }a,;0\cdot a=a\cdot 0=0$.	Same as semiring.
Additive inverses	Not required	Required

Note on terminology:

• Some authors require a multiplicative identity $1$ in both structures; others call a structure without $1$ a “hemiring” (for semirings without identity) or a “rng” (for rings without unity).
• In this summary we assume both have $1$, unless noted otherwise.

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2. Key Differences

1. Additive inverses

   • Ring: Every element $a$ has an inverse $-a$ so that $a+(-a)=0$.
   • Semiring: May lack inverses; subtraction need not be defined.

2. Examples

   • Semiring but not Ring:

     • $\mathbb{N}=0,1,2,\dots$ with usual $+$ and $\times$.
     • The Boolean semiring $0,1$ with $+$ = OR, $\cdot$ = AND.

   • Ring:

     • $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$.
     • Matrix rings $M_n(\mathbb{R})$.

3. Subtraction vs. cancellation

   • In rings you can “cancel” addition using inverses: from $a+x=a+y$ infer $x=y$.
   • In semirings, one often works with cancellative semirings to get a weak form of cancellation, but it’s not automatic.

4. Applications

   • Semirings:

     • Automata theory & formal languages (e.g.\ shortest-path semirings).
     • Tropical geometry (min-plus semiring).
     • Quantitative program analysis (probabilities, costs).

   • Rings:

     • Core algebra and number theory.
     • Algebraic geometry, homological algebra.

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3. Examples in More Detail

1. Natural numbers $(\mathbb{N},+,\times )$

   • Satisfies all semiring axioms: $0$ is additive identity, $1$ is multiplicative identity.
   • Fails ring axioms because there is no $n\in \mathbb{N}$ with $3+n=0$.

2. Tropical semiring

   • Underlying set $\mathbb{R}\cup \mathrm{\infty }$.
   • “Addition” is $min$, “multiplication” is $+$.
   • Used for shortest-path computations (Dijkstra, Bellman–Ford).

3. Boolean semiring $0,1$

   • $a+b=max(a,b)$ (logical OR), $a\cdot b=min(a,b)$ (logical AND).
   • Fundamental in logic, circuit design, and automata.

4. Integer ring $(\mathbb{Z},+,\times )$

   • Every integer has an additive inverse.
   • Ring of central importance in algebra and number theory.

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4. When to Use Which?

• Use a semiring when only nonnegative (or otherwise “one-sided”) quantities make sense and subtraction would be inappropriate or undefined.
• Use a ring when a full-fledged additive group structure (including negatives) is needed for constructions like modules, ideals, and homological tools.

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Further Reading

• Golan, J. S. Semirings and their Applications (Kluwer, 1999)
• Dummit & Foote, Abstract Algebra (especially the sections on rings and modules)

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By focusing on the presence or absence of additive inverses, you can quickly tell whether an algebraic system is a semiring or a ring—and choose the right framework for your application.

Groups vs rings

Groups vs rings

The concepts of groups and rings come from abstract algebra, and both are sets equipped with operations that satisfy certain rules. Here's a direct comparison to help clarify the difference:

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🔷 GROUPS

A group is a set with one binary operation (usually denoted as · or +) that satisfies the following properties:

1. Closure: If $a,b\in G$, then $a\cdot b\in G$.
2. Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity Element: There exists an element $e\in G$ such that $a\cdot e=e\cdot a=a$ for all $a\in G$.
4. Inverses: For each $a\in G$, there exists $a^{-1}\in G$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$.

• If $\cdot$ is commutative, i.e., $a\cdot b=b\cdot a$, the group is called an abelian group.
• Example: $(\mathbb{Z},+)$ is an abelian group.

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🔷 RINGS

A ring is a set equipped with two binary operations, usually addition and multiplication, satisfying:

1. The set is an abelian group under addition:

• Closure, associativity, identity (0), and additive inverses.
• Commutativity of addition.

2. The set is a semigroup under multiplication:

• Closure and associativity.
• (Not all rings require a multiplicative identity.)

3. Distributive laws:

• $a\cdot (b+c)=a\cdot b+a\cdot c$

• $(a+b)\cdot c=a\cdot c+b\cdot c$

• If multiplication is commutative, it's a commutative ring.

• If there is a multiplicative identity (1), it's a ring with unity.

• If every nonzero element has a multiplicative inverse, it's a division ring.

• If it's a commutative division ring, it's a field.

• Example: $(\mathbb{Z},+,\cdot )$ is a commutative ring with unity, but not a field (since not every nonzero integer has a multiplicative inverse in $\mathbb{Z}$).

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🔁 Summary Table

Feature	Group	Ring
Operations	One	Two (addition and multiplication)
Structure	One binary operation forming a group	Additive group + multiplicative semigroup
Identity Element	Required	Additive identity required; multiplicative identity optional
Inverses	Required for the operation	Required only for addition
Commutativity	Optional	Addition must be commutative
Examples	$(\mathbb{Z},+)$, $({\mathbb{R}}^{\ast },\cdot )$	$\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$

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If you'd like examples or help understanding a specific type (e.g. fields, division rings, or how groups/rings appear in real-world math), just let me know.

Homology and cohomology

What is homology and also cohomology?

Homology and cohomology are two foundational concepts in algebraic topology, a field of mathematics that studies topological spaces using algebraic tools. Here's a clear breakdown of what each one means and how they differ:

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Homology

Homology is a way to measure the "holes" in a topological space.

Intuition:

• Imagine different-dimensional "holes":

  • 0-dimensional: components (islands).
  • 1-dimensional: loops or holes (like the hole in a donut).
  • 2-dimensional: voids (like the hollow inside a sphere).

Idea:

Homology assigns a sequence of abelian groups (or vector spaces) $H_0,H_1,H_2,\dots$ to a topological space:

• $H_0$: captures connected components.
• $H_1$: captures loops (1D holes).
• $H_2$: captures voids (2D holes).
• And so on.

Example:

• A circle $S^1$:

  • $H_0(S^1)\cong \mathbb{Z}$ (1 connected component)
  • $H_1(S^1)\cong \mathbb{Z}$ (1 loop)
  • $H_n(S^1)=0$ for $n\ge 2$

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Cohomology

Cohomology is a theory closely related to homology but works in a dual way.

Intuition:

Cohomology still detects "holes," but it also gives a way to "probe" the space using functions, and can encode richer algebraic structure.

Key Differences from Homology:

• Cohomology groups are written as $H^0,H^1,H^2,\dots$ (note the superscript).
• Cohomology has a ring structure via the cup product, allowing interaction between cohomology classes.

Purpose:

• Offers more structure than homology (e.g., operations between cohomology classes).
• Useful in differential geometry, physics, and algebraic geometry.

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Comparison Table

Feature	Homology	Cohomology
Notation	$H_n(X)$	$H^n(X)$
Measures	n-dimensional holes	n-dimensional "cocycles"
Algebra	Abelian group	Abelian group + ring structure
Tools	Chains, cycles, boundaries	Cochains, cocycles, coboundaries
Duality	Direct computation	Dual of homology (in some sense)

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Why Two Theories?

• They are closely related but highlight different aspects.
• Cohomology often allows for more refined analysis.
• Many deep theorems (like Poincaré duality) relate the two.

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Let me know if you’d like an example computed, or a more visual explanation.

Euclidean algorithm halts with all-and-only commensurable inputs

show: the euclidean algorithm halts if, and only if, it is given commensurable quantities

Let $a,b>0$ be two magnitudes (or lengths). “Commensurable” means there is a third magnitude $d>0$ and positive integers $m,n$ with

$$a=md,b=nd,$$

i.e., $a$ and $b$ are integer multiples of a common measure $d$. The Euclidean algorithm (in the modern form using division with remainder, or equivalently the subtractive form) produces a sequence of remainders

$$r_0:=a,r_1:=b,r_{i+1}:=r_{i-1}mod{r}_i,$$

and halts exactly when some $r_k$ divides $r_{k-1}$ giving $r_{k+1}=0$; the last nonzero remainder is then a common measure.

We prove the equivalence:

(⇒) If the Euclidean algorithm halts on $a,b$, then $a$ and $b$ are commensurable. Suppose it halts in finitely many steps with last nonzero remainder $r_k$, so that $r_{k-1}=q_k{r}_k$ and the next remainder is zero. By the division relations backwards,

$$r_k\mid r_{k-1},r_k\mid r_{k-2},\dots ,r_k\mid r_1=b,\text{and }{r}_k\mid r_0=a.$$

Thus $a=M{r}_k,b=N{r}_k$ with integers $M,N$; so $r_k$ is a common measure and $a,b$ are commensurable.

(⇐) If $a$ and $b$ are commensurable, say $a=md,b=nd$ with $m,n\in {\mathbb{Z}}_{>0}$, then the Euclidean algorithm on $a,b$ reduces to the usual integer Euclidean algorithm on $m,n$ with unit $d$: each division step

$$r_{i-1}=q_i{r}_i+r_{i+1}$$

when scaled by $d^{-1}$ becomes an integer division among the corresponding integers. Since the Euclidean algorithm on positive integers $m,n$ terminates in finitely many steps (the remainders form a strictly decreasing sequence of nonnegative integers), the algorithm on $a,b$ also terminates—yielding as last nonzero remainder $(gcd(m,n))d$, a common measure.

Therefore the Euclidean algorithm halts on $a,b$ if and only if $a$ and $b$ are commensurable. ∎

Viterbi semiring intro

Explain the Viterbi semiring

The Viterbi semiring is the algebraic structure underlying the Viterbi algorithm’s “max-product” computations — it formalizes how you score and select the single best path (most probable sequence) in models like Hidden Markov Models (HMMs) or weighted finite-state automata.

Core idea

A semiring is a set $S$ equipped with two binary operations:

• $\oplus$ (an “addition”-like operation, here used to choose between alternatives),
• $\otimes$ (a “multiplication”-like operation, here used to accumulate scores along a path),

along with identity elements $\mathbf{0}$ for $\oplus$ and $\mathbf{1}$ for $\otimes$, such that:

• $\oplus$ is associative and commutative, with identity $\mathbf{0}$,
• $\otimes$ is associative, with identity $\mathbf{1}$,
• $\otimes$ distributes over $\oplus$: $a\otimes (b\oplus c)=(a\otimes b)\oplus (a\otimes c)$,
• $\mathbf{0}$ is absorbing for $\otimes$: $\mathbf{0}\otimes a=\mathbf{0}$.

The Viterbi semiring for probabilities is typically:

• Domain: $[0,1]$ (or, for numerical stability, log-space: $\mathbb{R}\cup \{-\mathrm{\infty }\}$),
• $\oplus =max$ (choose the most probable alternative),
• $\otimes =\times$ (multiply probabilities along a path),
• Identities: $\mathbf{0}=0$ (since $max(x,0)=x$ for $x\in [0,1]$), $\mathbf{1}=1$.

In log-space, this becomes the max-plus semiring:

• Domain: $\mathbb{R}\cup \{-\mathrm{\infty }\}$,
• $\oplus =max$,
• $\otimes =+$,
• Identities: $\mathbf{0}=-\mathrm{\infty }$, $\mathbf{1}=0$.

Switching to log-space turns products into sums (safer numerically) and keeps the “best path” selection as a max.

Connection to the Viterbi algorithm

The Viterbi algorithm performs dynamic programming over this semiring: at each time step $t$, for each state $j$, it computes

$${\delta }_t(j)=\underset{i}{⨁}[{\delta }_{t-1}(i)\otimes a_{i,j}]\otimes b_j(o_t)$$

which, in the usual (probability) notation, is

$${\delta }_t(j)=\underset{i}{max}({\delta }_{t-1}(i)\cdot a_{i,j})\cdot b_j(o_t),$$

where:

• ${\delta }_t(j)$ is the max probability of any path ending in state $j$ after observing $o_1,\dots ,o_t$,
• $a_{i,j}$ is the transition probability from state $i$ to $j$,
• $b_j(o_t)$ is the emission probability of observation $o_t$ from state $j$.

In log-space:

$$\log {\delta }_t(j)=\underset{i}{max}(\log {\delta }_{t-1}(i)+\log a_{i,j})+\log b_j(o_t).$$

Backpointers record which predecessor $i$ achieved the $max$, so the single best full path can be reconstructed.

Small concrete example

HMM with two states $A,B$:

• Initial probs: ${\pi }_A=0.6,{\pi }_B=0.4$

• Transitions: $A\to A=0.7,A\to B=0.3;B\to A=0.4,B\to B=0.6$

• Emissions for observation sequence $[x,y]$:

  • From $A$: $P(x)=0.5,P(y)=0.5$
  • From $B$: $P(x)=0.1,P(y)=0.9$

Step 1 (observe $x$):

$${\delta }_1(A)=0.6\times 0.5=0.3,{\delta }_1(B)=0.4\times 0.1=0.04.$$

Step 2 (observe $y$):

$$\begin{gathered} {\delta }_2(A)=max(0.3\times 0.7,0.04\times 0.4)\times 0.5=max(0.21,0.016)\times 0.5=0.105, \\ {\delta }_2(B)=max(0.3\times 0.3,0.04\times 0.6)\times 0.9=max(0.09,0.024)\times 0.9=0.081. \end{gathered}$$

Final best state is $A$ with probability $0.105$; backpointers give path $A\to A$.

Contrast: sum-product semiring

The sum-product semiring (used for computing total probability or marginals) has:

• $\oplus =+$,
• $\otimes =\times$,
• Identities: $\mathbf{0}=0,\mathbf{1}=1$.

There, you sum over paths instead of taking the max, so you’re aggregating total mass rather than picking one best path. In log-space that requires log-sum-exp for $\oplus$, which is more expensive than $max$.

Why thinking in semirings helps

Many dynamic programming algorithms (Viterbi, shortest path, parsing, belief propagation variants) can be seen as instances of the generalized semiring framework: replace $\oplus$ and $\otimes$ to get different behaviors (e.g., highest weight path vs. total weight, counting paths, etc.). The Viterbi semiring specializes this to “best path under multiplicative scoring.”

Properties summary

• Associativity / commutativity: $max$ is commutative/associative; multiplication (or addition in log-space) is associative.
• Distributivity: $a\otimes max(b,c)=max(a\otimes b,a\otimes c)$ holds.
• No additive inverse: It’s a semiring, not a ring — there is no operation to “undo” a $max$.
• Used for decoding: In speech recognition, bioinformatics, NLP (e.g., Viterbi taggers, parse scoring), etc.

If you want, I can also show how the Viterbi semiring fits into a generic semiring-based code template or how to generalize it to other semirings. Would you like that?

Linear equation systems have 0 ∨ 1 ∨ ∞ solutions

Show that a system of linear equations either has no solution, or exactly one solution, or infinitely many solutions.

Let $A\in {\mathbb{R}}^{m\times n}$, $b\in {\mathbb{R}}^m$, and consider the linear system

$$Ax=b.$$

Let $S=\{x\in {\mathbb{R}}^n:Ax=b\}$ be its solution set. There are three possibilities:

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1. No solution. The system is inconsistent (e.g., row-reduction of the augmented matrix $[A\mid b]$ produces a row $[0\cdots 0\mid c]$ with $c\ne 0$). Then $S=\varnothing$.

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2. At least one solution. Suppose $x_0$ is a solution: $A{x}_0=b$. Define the homogeneous solution space

$$H=\{h\in {\mathbb{R}}^n:Ah=0\}$$

(which is a subspace of ${\mathbb{R}}^n$). Then for any $h\in H$, $A(x_0+h)=A{x}_0+Ah=b+0=b$, so $x_0+H\subseteq S$. Conversely, if $x\in S$, then $A(x-x_0)=Ax-A{x}_0=b-b=0$, so $x-x_0\in H$ and $x=x_0+h$ for some $h\in H$. Hence

$$S=x_0+H,$$

i.e., the solution set is an affine subspace parallel to the homogeneous solution space.

Now:

• If $H=\{0\}$, then $S=\{x_0\}$: exactly one solution.
• If $H$ contains a nonzero vector $h\ne 0$, then for each real scalar $t$, $x_0+th\in S$. If $t_1\ne t_2$, then $(x_0+t_{1}h)-(x_0+t_{2}h)=(t_1-t_2)h\ne 0$, so these are all distinct. Thus there are infinitely many solutions.

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Conclusion: Over $\mathbb{R}$ (or any infinite field), a linear system has either no solution, exactly one solution, or infinitely many solutions—no other cardinalities occur.

Example:

• $x+y=1,x+y=2$ is inconsistent ⇒ no solution.
• $x+y=1,x-y=0$ has a unique solution $x=y=\frac{1}{2}$.
• $x+y=2,2x+2y=4$ is consistent but dependent: $S=\{(t,2-t):t\in \mathbb{R}\}$ ⇒ infinitely many solutions.

(Note: over a finite field, the “infinitely many” case becomes “more than one” but still finite; the trichotomy above assumes the underlying field is infinite, as in $\mathbb{R}$.)

Quasigroups

Explain quasigroups

A quasigroup is an algebraic structure $(Q,\ast )$ consisting of a set $Q$ with a binary operation $\ast$ such that for every $a,b\in Q$ the equations

• $a\ast x=b$, and
• $y\ast a=b$

have unique solutions $x,y\in Q$. In other words, “division” is always possible and yields a single result on both sides. This is equivalent to saying that the Cayley (multiplication) table of a finite quasigroup is a Latin square: each element appears exactly once in each row and each column.

You can define two division operations:

• Left division: $a\mathrm{\setminus }b$ is the unique $x$ with $a\ast x=b$.
• Right division: $b/a$ is the unique $y$ with $y\ast a=b$.

These satisfy identities like

$$a\ast (a\mathrm{\setminus }b)=b,(b/a)\ast a=b,a\mathrm{\setminus }(a\ast b)=b,(b\ast a)/a=b.$$

Basic properties:

• Cancellation: From uniqueness of solutions, left and right cancellation hold: if $a\ast b=a\ast c$, then $b=c$; similarly $b\ast a=c\ast a$ implies $b=c$.
• Not necessarily associative: The operation need not satisfy $(a\ast b)\ast c=a\ast (b\ast c)$.
• No need for identity: A quasigroup may or may not have an identity element.

A loop is a quasigroup that does have a two-sided identity element $e$ such that $e\ast a=a=a\ast e$ for all $a$. So every group is a loop (and hence a quasigroup) with associativity; but loops in general need not be associative.

Examples:

1. Group example (associative loop): $(\mathbb{Z},+)$ with addition is a quasigroup (in fact a group) because for any $a,b$ the equations $a+x=b$ and $y+a=b$ have unique solutions $x=b-a$, $y=b-a$. Identity $0$ exists and associativity holds.

2. Non-associative quasigroup (no identity): Take $Q={\mathbb{Z}}_3=\{0,1,2\}$ and define $a\ast b=a-b(\mathrm{mod}3)$. Then for any $a,b$, $a\ast x=b$ has unique solution $x=a-b$, and similarly on the left. There is no single element acting as a two-sided identity for all $a$, and the operation is not associative in general. So $({\mathbb{Z}}_3,a\ast b=a-b)$ is a quasigroup but not a loop or group.

3. Latin square example: Let $Q=\{1,2,3\}$ with operation given by the Latin square

$$\begin{array}{cccc}\ast & 1& 2& 3\\ 1& 2& 3& 1\\ 2& 3& 1& 2\\ 3& 1& 2& 3\end{array}$$

This defines a quasigroup structure (every element appears once per row/column). It’s nontrivial to check associativity; this particular one is not necessarily associative, so it’s a non-associative quasigroup.

Special notions:

• Isotopy: Two quasigroups $(Q,\ast )$ and $(Q,\circ )$ are isotopic if there exist bijections $\alpha ,\beta ,\gamma :Q\to Q$ such that for all $x,y$,

  $$\alpha (x)\circ \beta (y)=\gamma (x\ast y).$$

  Isotopy generalizes isomorphism; many properties (like being a Latin square) are preserved under isotopy.

• Parastrophes: From a quasigroup one can derive up to six related quasigroups by permuting arguments and solving for different variables—these are the parastrophes (e.g., exchanging the roles of left/right division).

• Moufang and Bol loops: These are loops (quasigroups with identity) that satisfy extra identities weaker than full associativity, giving them group-like behavior in certain contexts.

Key theorem: If a quasigroup is associative, then it has a two-sided identity and inverses, so it is a group. Thus “associative quasigroup” $⟺$ group.

Applications: Quasigroups and their finite versions (Latin squares) appear in design theory (e.g., experimental design), error-correcting codes, cryptography (e.g., quasigroup string transformations / stream ciphers), and combinatorial constructions.

If you’d like, I can show more concrete examples (e.g., construct all quasigroups of small order), explore loops like Moufang loops, or prove the associative-quasigroup-is-group fact. Would you prefer a particular direction?

Is this different from having inverses?

Yes. The quasigroup/division property is weaker and more general than “having inverses” in the usual group sense.

• In a quasigroup you can always solve $a\ast x=b$ and $y\ast a=b$ uniquely for $x,y$; this gives you left- and right-“division” operators, but there need not be any distinguished identity element. So you don’t get a global inverse of an element because “inverse” is only defined relative to an identity.

• In a loop (a quasigroup with a two-sided identity $e$) you can define for each element $a$:

  • A left inverse $a^L$ such that $a^L\ast a=e$,
  • A right inverse $a^R$ such that $a\ast a^R=e$. These exist uniquely, but in a general loop $a^L$ and $a^R$ might be different. If they coincide and behave well (e.g., $(a^{-1}{)}^{-1}=a$, $(ab{)}^{-1}=b^{-1}{a}^{-1}$), the loop is said to have the inverse property (many Moufang loops do).

• In a group (an associative loop) there is an identity and each element has a two-sided inverse $a^{-1}$ with $a^{-1}\ast a=a\ast a^{-1}=e$; associativity makes these inverses behave in the familiar algebraic way.

So: quasigroup division gives you the ability to “undo” $a\ast$ or $\ast a$ in isolation, but without an identity you can’t talk about the inverse of $a$ itself. Having a (two-sided) inverse is a stronger structure that requires at least a loop, and in full generality (as in groups) also associativity.

Examples of theory development according to Halmos

Paul Halmos said:

It has often happened that a theory designed originally as a tool for the study of a physical problem came subsequently to have purely mathematical interest. When that happens, the theory is usually generalized way beyond the point needed for applications, the generalizations make contact with other theories (frequently in completely unexpected directions), and the subject becomes established as a new part of pure mathematics. The part of pure mathematics so created does not (and need not) pretend to solve the physical problem from which it arises; it must stand or fall on its own merits.

Give examples of this.

Below are some well-loved cases where a tool built for a concrete physical problem was abstracted, stretched, and woven so deeply into mainstream mathematics that it now flourishes with little—or no—regard for the original application.

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1. Calculus ⇒ Real & Complex Analysis, Measure Theory

Physics seed. Newton and Leibniz invented the differential and integral calculus to describe planetary motion and falling bodies. Mathematical flowering. Over the next three centuries questions about rigor, convergence, and “what is a function?” spawned ε-δ analysis, measure theory, Lebesgue integration, and modern functional analysis. Today vast stretches of real and complex analysis have no direct tie to mechanics, yet the language of limits, continuity, and integration they introduced underpins almost every branch of pure mathematics.

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2. Fourier’s Heat Equation ⇒ Harmonic Analysis & Representation Theory

Physics seed. In 1807 Fourier decomposed temperature distributions into sine waves to solve the heat equation. Mathematical flowering. Generalizing those trigonometric series to arbitrary groups, spaces, and operators led to harmonic analysis, abstract Plancherel theorems, and deep links to group representation theory and number theory (e.g., modular forms, automorphic representations). The toolbox now serves problems that have nothing to do with heat flow.

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3. Hilbert Space & Operators ⇒ Modern Functional Analysis and Operator Algebras

Physics seed. Integral-equation formulations of diffraction and quantum mechanics in the early 1900s forced mathematicians (notably Hilbert) to treat infinite-dimensional inner-product spaces systematically. Mathematical flowering. Banach’s and von Neumann’s generalizations (Banach spaces, C*- and von Neumann algebras, spectral theory) turned functional analysis into its own kingdom, fundamental to fields ranging from PDE to ergodic theory and pure operator algebra classification, far removed from Schrödinger’s equation.

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4. Dirac Delta ⇒ Distribution Theory

Physics seed. Dirac introduced the “δ-function’’ to describe point charges and quantum measurements—an object that made physical sense but defied classical function definitions. Mathematical flowering. Schwartz formalized distributions (generalized functions) in the 1940s, creating a robust framework for linear PDEs, microlocal analysis, and the modern theory of Sobolev spaces. Many distribution-theoretic developments concern abstract regularity and propagation of singularities, with no physical context required.

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5. Hamiltonian Mechanics ⇒ Symplectic Geometry & Topology

Physics seed. Hamilton packaged classical mechanics in the 1830s using conjugate position–momentum coordinates. Mathematical flowering. Treating the 2n-dimensional phase space abstractly gave rise to symplectic manifolds. Gromov’s 1985 pseudoholomorphic-curve theory, Floer homology, and mirror symmetry connected symplectic geometry to topology, algebraic geometry, and category theory—fields where “mass” and “momentum” never appear.

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6. Kelvin’s Vortex Atoms ⇒ Knot Theory & Low-Dimensional Topology

Physics seed. Lord Kelvin (1867) imagined atoms as knotted vortices in the ether, motivating Tait to tabulate knots. Mathematical flowering. Though the ether died, the knot tables survived. Reidemeister moves, Alexander, Jones, and HOMFLY-PT polynomials, plus 3-manifold invariants such as Chern–Simons–Witten theory now occupy a central place in topology, representation theory, and even quantum computing.

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7. Yang–Mills Gauge Fields ⇒ 4-Manifold Topology & Geometric Analysis

Physics seed. 1950s non-abelian gauge theory described the strong and weak nuclear forces. Mathematical flowering. Donaldson (1983) used the Yang–Mills moduli space to produce exotic smooth structures on ℝ⁴; Seiberg–Witten theory refined these techniques. The resulting revolution in differential topology and low-dimensional geometry persists regardless of particle physics.

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8. Random Matrix Theory (Wigner’s Nuclear Physics) ⇒ Number Theory & Combinatorics

Physics seed. Wigner modeled heavy-nucleus energy levels with random symmetric matrices in the 1950s. Mathematical flowering. The statistics of eigenvalues (level spacing, Tracy–Widom laws) were abstracted and now illuminate the zeros of the Riemann zeta function, growth of Young diagrams, and universality questions in combinatorics—no neutron spectra in sight.

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9. Statistical Mechanics ⇒ Ergodic Theory & Dynamical Systems

Physics seed. Boltzmann and Gibbs created ensembles to explain gas thermodynamics. Mathematical flowering. Abstract notions of measure-preserving transformations, entropy, and mixing evolved into ergodic theory. Today, rigidity theorems for group actions or homogeneous flows live squarely in pure mathematics, even when the systems studied have no atoms at all.

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10. Quantum Integrable Models ⇒ Quantum Groups & Knot Invariants

Physics seed. Exactly solvable 1-D quantum spin chains used R-matrices satisfying the Yang–Baxter equation. Mathematical flowering. Drinfeld and Jimbo abstracted that combinatorial property into “quantum groups,” which now influence category theory, low-dimensional topology, and representation theory—the physical spins have faded from view.

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Halmos’s cycle—applied tool → wild generalization → unexpected bridges → autonomous theory—recurs throughout mathematics. The stories above show how questions born in the laboratory or observatory can end up enriching the pure landscape in ways their inventors could never have predicted.

Prime-order group powers: vector spaces

powers of groups of prime order are vector spaces

Yes, that's correct — groups of prime order have an important connection to vector spaces, under the right interpretation.

Key Idea:

If $G$ is a group of prime order $p$, then:

• $G$ is a cyclic group (since all groups of prime order are cyclic),
• Every element $g\in G$ satisfies $g^p=e$, the identity element,
• If $G$ is also abelian and we consider it as a module over ${\mathbb{F}}_p$, then it becomes a vector space over ${\mathbb{F}}_p$.

---

Explanation:

Let’s formalize it:

1. Group of Prime Order:

Let $G$ be a group with order $p$, a prime number.

• Then $G$ is cyclic: $G\cong \mathbb{Z}/p\mathbb{Z}$,
• It is abelian, because all cyclic groups are abelian.

2. Powers and Vector Space Structure:

If you now look at a finite abelian group $G$ where every element has order dividing a prime $p$, i.e., for all $g\in G$, $g^p=e$, then:

• $G$ is a ${\mathbb{F}}_p$-vector space.

Why?

Because we can define scalar multiplication by elements of ${\mathbb{F}}_p$ as:

$$a\cdot g=g^a$$

This satisfies the vector space axioms because exponentiation respects the group operation.

Thus:

Any abelian group $G$ where every element has order $p$ (a prime) can be regarded as a vector space over ${\mathbb{F}}_p$.

This includes:

• Any group of order $p$,
• More generally, any abelian $p$-group of exponent $p$.

---

Examples:

• $\mathbb{Z}/p\mathbb{Z}\cong {\mathbb{F}}_p$: 1-dimensional vector space over ${\mathbb{F}}_p$,
• $(\mathbb{Z}/p\mathbb{Z}{)}^n\cong {\mathbb{F}}_p^n$: n-dimensional vector space over ${\mathbb{F}}_p$.

---

Conclusion:

So yes — the powers (i.e., elements raised to integers) in abelian groups of prime order, or more generally abelian $p$-groups of exponent $p$, behave like elements of a vector space over the finite field ${\mathbb{F}}_p$.

Associative quasigroups: They are groups

Explain associative quasigroups

Sure. Here’s the picture in full:

1. Definitions

• A quasigroup $(Q,\ast )$ is a set with a binary operation $\ast$ such that for every $a,b\in Q$ the equations $a\ast x=b$ and $y\ast a=b$ have unique solutions $x,y\in Q$. Equivalently, left and right multiplication by any fixed element are bijections. (Wikipedia)

• A loop is a quasigroup that has an identity element $e$ with $e\ast a=a\ast e=a$ for all $a$. (Wikipedia)

• An associative quasigroup is a quasigroup whose operation is associative: $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in Q$. (groupprops.subwiki.org)

2. Main fact

Any nonempty associative quasigroup is a group. In other words, associativity together with the quasigroup division property forces the existence of a two-sided identity and two-sided inverses, so $(Q,\ast )$ satisfies all the group axioms. Conversely, every group is trivially an associative quasigroup. (groupprops.subwiki.org, Wikipedia)

(Minor technical note: the empty set can be vacuously an associative quasigroup under some conventions, but it lacks an identity element and so is not a group in the usual sense. The standard theorem refers to nonempty associative quasigroups. ) (Mathematics Stack Exchange)

3. Sketch of the proof (from the standard argument): (groupprops.subwiki.org)

Let $(Q,\ast )$ be an associative quasigroup and pick some $a\in Q$ (so $Q$ is assumed nonempty).

• By the quasigroup property, there is a unique $e\in Q$ with $a\ast e=a$. For an arbitrary $b\in Q$, pick $y$ with $y\ast a=b$; then

  $$b\ast e=(y\ast a)\ast e=y\ast (a\ast e)=y\ast a=b$$

  using associativity. Thus $e$ is a right identity for all elements. Likewise one constructs a left identity, and uniqueness of left/right neutral elements forces them to coincide, giving a two-sided identity $e$.

• Now with identity $e$ in hand, for any $a\in Q$ the quasigroup property gives $x,y$ such that $a\ast x=e$ and $y\ast a=e$. Associativity implies any left and right inverse coincide, so each element has a two-sided inverse.

Hence $Q$ has an identity and every element is invertible, and the operation is associative: it is a group. (groupprops.subwiki.org)

4. Consequences / interpretation

So “associative quasigroup” (with nonempty underlying set) is just another way to axiomatize a group: one can define a group as an associative quasigroup, without separately postulating an identity and inverses. Historically some early definitions of groups were phrased that way. (groupprops.subwiki.org)

5. Examples

• Any group $(G,\cdot )$ is an associative quasigroup because it has associativity and the division property (invertibility gives unique solutions to $a\ast x=b$ and $y\ast a=b$). (Wikipedia)
• There are no new “non-group” examples: if you assume associativity in a quasigroup, you’re forced into a group structure. (groupprops.subwiki.org)

If you’d like, I can also give variants (e.g., what weaker associativity-like identities still force a quasigroup to be a loop/group) or examples of nonassociative quasigroups to contrast. Would you like that?

Vector space from field

Show that if ${\mathbb{F}}_q$ is a field with $q$ elements, then ${\mathbb{F}}_q^n$ is a vector space of dimension $n$ over ${\mathbb{F}}_q$.

Why ${\mathbb{F}}_q^n$ is a vector space over ${\mathbb{F}}_q$ Take the set of all $n$-tuples of elements of ${\mathbb{F}}_q$:

$${\mathbb{F}}_q^n=\{(a_1,\dots ,a_n)\mid a_i\in {\mathbb{F}}_q\}.$$

Define the usual component-wise operations

• Addition: $(a_1,\dots ,a_n)+(b_1,\dots ,b_n)=(a_1+b_1,\dots ,a_n+b_n)$.
• Scalar multiplication: for $\lambda \in {\mathbb{F}}_q$, $\lambda (a_1,\dots ,a_n)=(\lambda a_1,\dots ,\lambda a_n)$.

Because ${\mathbb{F}}_q$ is already a field, its addition “$+$” and multiplication “$\cdot$” satisfy all field axioms. Applying them component-wise:

• Closure, associativity, commutativity, additive identity $0=(0,\dots ,0)$, and additive inverses $-(a_1,\dots ,a_n)=(-a_1,\dots ,-a_n)$ follow immediately from the corresponding properties in ${\mathbb{F}}_q$.
• For scalar multiplication, distributivity, associativity, and the identity element $1\in {\mathbb{F}}_q$ also transfer component-wise.

Hence the standard 8 vector-space axioms hold, so ${\mathbb{F}}_q^n$ is indeed a vector space over ${\mathbb{F}}_q$.

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A basis and the dimension

Consider the $n$ canonical unit vectors

$$e_1=(1,0,\dots ,0),e_2=(0,1,0,\dots ,0),\dots ,e_n=(0,\dots ,0,1)\in {\mathbb{F}}_q^n.$$

• Spanning: Any vector $v=(a_1,\dots ,a_n)$ can be written

  $$v=a_1{e}_1+\cdots +a_n{e}_n,$$

  so $\{e_1,\dots ,e_n\}$ spans ${\mathbb{F}}_q^n$.

• Linear independence: If ${\lambda }_1{e}_1+\cdots +{\lambda }_n{e}_n=0$ then each coordinate is ${\lambda }_i=0$ in ${\mathbb{F}}_q$, so the only linear relation is the trivial one.

Therefore $\{e_1,\dots ,e_n\}$ is a basis. A basis has exactly $n$ vectors, so the dimension of ${\mathbb{F}}_q^n$ over ${\mathbb{F}}_q$ is $n$.

$$\boxed{{\displaystyle {\dim }_{{\mathbb{F}}_q}({\mathbb{F}}_q^n)=n.}}$$

Matrices: Abelian Group?

The set of matrices is an abelian group?

To determine whether a set of matrices forms an abelian group, we need to be more specific — for example:

1. Which set of matrices?

   • All square matrices?
   • Invertible matrices?
   • Diagonal matrices?
   • Symmetric matrices?
   • Upper/lower triangular matrices?
   • Over which field? (Real numbers ℝ, complex numbers ℂ, etc.)

2. Which operation?

   • Matrix addition or matrix multiplication?

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Let's go through common cases:

✅ Under matrix addition:

• The set of all n × n matrices over a field (like ℝ or ℂ) is an abelian group under addition:

  • Closure: Sum of two n × n matrices is an n × n matrix.
  • Associativity: Matrix addition is associative.
  • Identity: The zero matrix.
  • Inverses: Each matrix has an additive inverse (its negative).
  • Commutativity: Matrix addition is commutative.

✅ So yes, (ℝⁿˣⁿ, +) is an abelian group.

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❌ Under matrix multiplication:

• The set of all n × n matrices over a field is not a group under multiplication:

  • Not all matrices are invertible.
  • Matrix multiplication is not commutative in general.

❌ So it is not even a group, let alone an abelian one.

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✅ Invertible matrices under multiplication:

• The set of invertible n × n matrices (called GL(n, ℝ) or GL(n, ℂ)) forms a group under multiplication.

  • But it is not abelian for n > 1 (since multiplication is not commutative).

✅ Group? Yes ❌ Abelian? No (for n > 1)

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Summary:

Set of Matrices	Operation	Group?	Abelian?
All n × n matrices (ℝⁿˣⁿ)	Addition	Yes	Yes
All n × n matrices (ℝⁿˣⁿ)	Multiplication	No	No
Invertible n × n matrices (GL(n))	Multiplication	Yes	No
1 × 1 invertible matrices	Multiplication	Yes	Yes

Let me know which specific set and operation you're referring to if you want a more targeted answer.

Linear Difference Equations

What are linear difference equations?

Linear difference equations are equations that define a sequence recursively, meaning each term in the sequence is expressed as a linear combination of previous terms. They are the discrete-time analog of linear differential equations.

General Form

A linear difference equation of order $n$ has the form:

$$a_n{y}_{k+n}+a_{n-1}{y}_{k+n-1}+\cdots +a_1{y}_{k+1}+a_0{y}_k=f(k)$$

• $y_k$ is the sequence (dependent variable).
• $a_0,a_1,\dots ,a_n$ are constants (or sometimes functions of $k$).
• $f(k)$ is a known function (can be zero for homogeneous equations).

Types

1. Homogeneous linear difference equation: when $f(k)=0$.
2. Non-homogeneous: when $f(k)\ne 0$.
3. Constant-coefficient: when all $a_i$ are constants.
4. Variable-coefficient: when some $a_i$ depend on $k$.

Examples

1. First-order homogeneous:

   $$y_{k+1}-3y_k=0$$

   Solution: $y_k=C\cdot 3^k$

2. Second-order non-homogeneous:

   $$y_{k+2}-4y_{k+1}+4y_k=2^k$$

Applications

• Time series analysis (e.g., AR models)
• Population modeling
• Control theory
• Signal processing
• Numerical algorithms

Would you like help solving a specific example?

Equality is both symmetric and antisymmetric

"Equality is both symmetric and antisymmetric"

The statement "Equality is both symmetric and antisymmetric" is true, and it's a great example to understand these properties in the context of relations.

Let’s break it down:

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Equality as a Relation

Equality ( = ) is a relation defined on a set — say, the set of real numbers ℝ.

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Symmetric Relation

A relation $R$ is symmetric if:

If $aRb$, then $bRa$

For equality:

If $a=b$, then $b=a$

✅ This is clearly true, so equality is symmetric.

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Antisymmetric Relation

A relation $R$ is antisymmetric if:

If $aRb$ and $bRa$, then $a=b$

For equality:

If $a=b$ and $b=a$, then $a=b$

✅ This is also trivially true, because if both $a=b$ and $b=a$ hold, then obviously $a=b$

So equality is also antisymmetric.

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Conclusion

Equality is both symmetric and antisymmetric. This is possible because antisymmetry doesn’t prohibit symmetry — it just requires that if both directions hold, the elements must be equal, which they are in the case of equality.

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If you want a quick comparison:

Property	Does Equality Have It?	Why
Reflexive	✅ Yes	$a=a$ always holds
Symmetric	✅ Yes	$a=b\Rightarrow b=a$
Antisymmetric	✅ Yes	$a=b$ and $b=a\Rightarrow a=b$
Transitive	✅ Yes	$a=b$ and $b=c\Rightarrow a=c$

So equality is an example of a partial order and also an equivalence relation — in fact, it's the strongest form of relation: complete identity.

Monoid examples

Examples of monoids

A monoid is an algebraic structure with a single associative binary operation and an identity element. Formally, a monoid is a set $M$ with a binary operation $\cdot$ such that:

1. Closure: For all $a,b\in M$, $a\cdot b\in M$.
2. Associativity: For all $a,b,c\in M$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity: There exists an element $e\in M$ such that for all $a\in M$, $e\cdot a=a\cdot e=a$.

Here are some common examples of monoids:

---

1. Natural Numbers with Addition

• Set: ${\mathbb{N}}_0=\{0,1,2,\dots \}$
• Operation: Addition $(+)$
• Identity: 0

---

2. Strings with Concatenation

• Set: All strings over an alphabet ${\mathrm{\Sigma }}^{\ast }$
• Operation: String concatenation
• Identity: Empty string $\epsilon$

---

3. Integers with Multiplication

• Set: $\mathbb{Z}$
• Operation: Multiplication $(\cdot )$
• Identity: 1

---

4. Boolean Values with AND or OR

• Set: $\{\text{true},\text{false}\}$

  • With AND: Identity is true
  • With OR: Identity is false

---

5. Matrices with Matrix Multiplication

• Set: All $n\times n$ matrices over a field
• Operation: Matrix multiplication
• Identity: Identity matrix $I_n$

---

6. Functions from a Set to Itself (Endofunctions)

• Set: All functions $f:X\to X$
• Operation: Function composition
• Identity: Identity function ${\text{id}}_X$

---

7. Lists under Concatenation

• Set: Lists of elements (e.g., integers)
• Operation: Concatenation of lists
• Identity: Empty list []

---

Let me know if you want examples from programming (e.g., Haskell, Python), category theory, or more abstract algebra.

On the Primitive Term of Logistic

Tell me about "On the primitive term of logistic"

What is it?

• Title (original): O wyrazie pierwotnym logistyki
• Author: Alfred Tarski (then publishing under the Polish spelling Tajtelbaum-Tarski)
• Form: Doctoral dissertation defended at the University of Warsaw in 1924; first printed (in Polish) in Przegląd Filozoficzny 26 (1923); authoritative English translation in Tarski’s Logic, Semantics, Metamathematics (1956, 2nd ed. 1983) (Wikipedia, Google Books)
• Length & setting: Roughly 25 journal pages; written inside the vibrant Warsaw‐school tradition led by Leśniewski, Łukasiewicz and others.

The central problem

Tarski asks:

Can one truth-functional sign, together with the usual quantifiers, serve as the sole primitive basis of “logistic” (i.e. classical first-order logic)?*

At the time logicians already knew two single-connective solutions:

Solution	Primitive sign	Discovered by
Implication (⊃)	Bertrand Russell (1910)
Sheffer stroke (NAND / ↑ or NOR / ↓)	Henry Sheffer (1913)

Tarski’s twist is to try material equivalence (↔) instead. In his view the biconditional has two attractive features that the others lack:

1. Natural definitional shape – every definition can be phrased as an equivalence.
2. Symmetry & reflexivity – ↔ treats its arguments on a par, mirroring the idea that definitions connect two sides of an identity. (Internet Encyclopedia of Philosophy)

How he does it (sketch)

1. Fundamental Theorem – Tarski shows how to define conjunction (p ∧ q) using only ↔, variables and a single universal quantifier:

   p ∧ q   ≡   ∀f [ (p ↔ f(r)) ↔ (q ↔ f(r)) ]
   

   (Here f is a variable for unary propositional functions, r a variable.)

2. Cascade of definitions – Once ∧ is available, he derives ¬, ∨, →, ⊥, etc., in the standard style of Whitehead-Russell.

3. Metatheoretic reflection – He explores when such reductions are legitimate, noting they should not smuggle in new constants and should be compatible with any reasonable type theory (he cites a system of Leśniewski as an existence proof).

Why it matters

• Historical pivot – This was the only PhD ever supervised by Stanisław Leśniewski, and it marks the first major publication of the 22-year-old Tarski, foreshadowing his lifelong concern with definitions and metalogical economy. (Wikipedia)
• Economy of primitives – Tarski’s method fueled later work on “equivalential calculi,” systems whose sole connective is ↔. These became a laboratory for studying definability, functional completeness and replacement properties (today visible in algebraic logic and structural proof theory).
• Influence on protothetic – Leśniewski’s own generalized propositional logic (Protothetic) eventually adopted ↔ as its only sentential connective, a move explicitly justified by Tarski’s proof.
• Pre-echo of model theory – The attention to arbitrary interpretation of function variables anticipates Tarski’s later semantic conception of truth and his use of schematic functional parameters in model‐theoretic arguments.

Reading tips & modern connections

• Where to find it in English: pp. 1-23 of Logic, Semantics, Metamathematics (Hackett 1983); the translation preserves Tarski’s original notation but adds section headings. (Google Books)
• Notation quirks: Quantifiers are written with leading dots (·[x]), and ↔ is denoted ‘==’. If you map these to ∀ and ↔, the proofs are straightforward Hilbert-style derivations.
• Limitations: Using ↔ without quantifiers is not functionally complete; Tarski’s reduction is essentially second-order (it quantifies over propositional functions). Later results (Post, Meredith, Kalmar) sharpened the analysis of single-connective completeness inside pure propositional logic.
• Legacy: Ideas from the paper resurface in single-axiom systems (e.g., Meredith’s single-axiom for classical logic) and inform contemporary proof-compression techniques in automated reasoning.

One-sentence takeaway

Tarski’s On the Primitive Term of Logistic is a youthful yet remarkably sophisticated demonstration that the elegant biconditional ↔—paired only with quantification—suffices to generate the whole machinery of classical logic, inaugurating a research program on the logical and philosophical significance of our choice of primitive notions.

How to defeat Magneto

In the X Men universe, police have trouble stopping Magneto because he can manipulate the metal in any guns they use. Eventually they try to use guns made out of plastic but it's not great. Eventually it turns out he can manipulate the iron in people's blood. How do they even defeat him?

Magneto is one of the most powerful and complex villains in the X-Men universe, and defeating him is never straightforward because of the sheer scale of his magnetic and electromagnetic abilities. You're absolutely right that:

• Metal weapons are useless because he can control or dismantle them.
• Plastic weapons are a workaround, but they're generally impractical.
• He can manipulate iron in human blood, as dramatically shown in X2: X-Men United, making even biological systems vulnerable.

So... how do they ever beat him?

There are a few common strategies across comics, movies, and animated shows:

---

1. Psychic Attack

• Telepaths like Professor X or Jean Grey are among the few people who can truly challenge Magneto, especially if they catch him off guard.
• Magneto usually wears a helmet to block telepathy, but if the X-Men remove or bypass it, a powerful psychic can shut him down mentally.

Example: In the comics and movies, Charles Xavier has immobilized Magneto using telepathy — but only when Magneto's helmet is off.

---

2. Overwhelm Him with Numbers or Strategy

• Sometimes, a well-coordinated team effort from the X-Men overwhelms him.
• While Magneto is nearly unstoppable alone, he’s not omnipotent. Strategic attacks from multiple angles — especially from non-metallic mutants like Storm, Cyclops, or Nightcrawler — can sometimes work.

Example: In X-Men: The Animated Series, they often defeat Magneto through teamwork and misdirection, not brute force.

---

3. Use of Non-Metallic Tech or Environments

• Fights in metal-free environments (or in space, or underground) can neutralize some of his power.
• In some stories, specialized non-metal restraints, prisons, or mutant-suppressing technology are used against him.

---

4. Betrayal or Psychological Leverage

• Magneto is not purely evil — he has strong convictions about mutantkind, and sometimes, emotional appeals from his children (like Scarlet Witch or Quicksilver) or Xavier himself can sway or stall him.
• In some timelines, he even joins the X-Men out of necessity or guilt.

---

5. De-powered or Weakened

• In X-Men: The Last Stand (2006), Magneto is injected with a “cure” that removes his powers.
• In some comic arcs, he is temporarily depowered by technology, psychic assault, or his powers overloading.

---

So ultimately, defeating Magneto requires:

• Removing his helmet (if psychic attack is an option)
• Outsmarting him (traps, misdirection, tech)
• Teamwork
• Or moral/emotional appeals.

He’s like a boss fight with no easy win condition — every victory comes at a cost, and he almost always gets back up again.

FOL formulas as a ring

fol formulae as a ring under entailment

In logic, particularly in First-Order Logic (FOL), we can define structures that behave like algebraic rings under certain operations. While FOL formulas aren't literally a ring in the strict algebraic sense (like integers with addition and multiplication), there is an abstract analogy that treats formulas as forming an algebraic structure under logical entailment.

Here's an outline of how FOL formulae can be seen as forming a ring-like structure under entailment:

---

🔹 Preliminaries

Let’s consider:

• A set of first-order logic sentences: $\mathcal{L}$

• A consequence relation $\models$ (semantic entailment)

• The equivalence class $[\varphi ]$ of a formula $\varphi$, defined by:

  $$[\varphi ]=\{\psi \in \mathcal{L}\mid \varphi \models \psi \text{ and }\psi \models \varphi \}$$

  (This is logical equivalence: $\varphi \equiv \psi$)

This quotient structure $\mathcal{L}/\equiv$ consists of logical equivalence classes.

---

🔹 Ring-Like Operations

Define two operations on equivalence classes of formulas:

1. Addition-like (⊕):

   $$[\varphi ]\oplus [\psi ]:=[\varphi \leftrightarrow \psi ]$$

   This operation captures the symmetric difference of truth — a kind of logical XOR.

2. Multiplication-like (⊗):

   $$[\varphi ]\otimes [\psi ]:=[\varphi \wedge \psi ]$$

   This is conjunction of formulas.

These are well-defined on equivalence classes.

---

🔹 Ring Axioms (Mod 2 ring analogy)

Under these operations:

• $\oplus$ is commutative and associative
• $\otimes$ distributes over $\oplus$
• $[\mathrm{\perp }]$ (falsehood) acts as the additive identity
• $[\mathrm{\top }]$ (tautology) can serve as a multiplicative identity
• $[\varphi ]\oplus [\varphi ]=[\mathrm{\perp }]$ (each element is its own additive inverse — like in ${\mathbb{F}}_2$)

Thus, the set of logical equivalence classes forms a commutative ring (more specifically, a Boolean ring).

---

🔹 Summary: FOL Formulas as a Boolean Ring

• The set of FOL formulas modulo logical equivalence under entailment forms a Boolean ring.

• Operations are:

  • $\varphi \oplus \psi :=\varphi \leftrightarrow \psi$
  • $\varphi \otimes \psi :=\varphi \wedge \psi$

• This works in a quotient space $\mathcal{L}/\equiv$, not on raw formulas.

---

Would you like a concrete example, or an algebraic formalization of this construction?

Anonymous feedback benefits

Cate Hall says:

Seek real feedback

It’s hard to overstate how overpowered this one is. If you aren’t trying to get real feedback from people who know you, you’re cooking without tasting. This is, like, the lowest hanging fruit for self-improvement, but few people really try to pick it.

In many contexts, the way to get good feedback is to give people a way to provide it anonymously. Anything else creates friction by layering on social dynamics. To get honest feedback, you want to make it as comfortable as possible for people to give it. You also want to make it easy to find -- I have a link to my feedback form in my Twitter bio, and get a few comments a week through it.

I imagine resistance from some people on the grounds that anonymity frees people to be assholes, but in my experience they rarely are. 90% of what I get in my inbox is either nonsense or nice -- I get lots of “keep up the good work!” type messages. In over a year, I’ve gotten maybe two messages intended to hurt my feelings. Once in a while, I get a message that’s a gut punch because it calls my attention to a real issue -- but then I get over it and get to work fixing the issue. (I wish people gave me more feedback -- I had to learn about my uptalk from YouTube comments.)

I don’t fully get it, please develop her argument a bit more. What are some kinds of people who might not want to criticize me to my face but might do so anonymously? I can only think of people who don’t really know me well, since then I’d question that they’re qualified to criticize me, but that’s a point against allowing anonymous criticism, not in favor.

Cate Hall’s point is that anonymity mainly removes the social and power-dynamic costs that keep people who actually know you from being fully candid. Below are several groups that often do have useful insight but will sometimes bite their tongues unless you give them a friction-free, anonymous channel:

Who might stay silent?	What holds them back face-to-face?	Why their feedback still matters
Direct reports / junior teammates	Fear of retaliation, stalled promotions, or being labelled “not a team player.” In most orgs the power imbalance is real.	They see your leadership habits daily; their vantage point is uniquely valuable for improving management style. Research on employee surveys shows anonymity encourages “the silent” to speak up candidly (business.com).
Peers & cross-functional colleagues	Social-capital risk: honest critique could strain a collaborative relationship or expose them to reciprocal criticism. People default to “polite but vague.”	They understand your work context and inter-dependencies better than outsiders; you learn how you’re impacting partner teams.
Mentors, higher-status contacts, or senior execs	It can feel awkward—or politically unwise—to punch down with direct criticism, especially if you’re junior or outside someone’s reporting line.	Their feedback is often the most experienced and strategic, but they have to feel 100 % safe to deliver a “hard truth.”
Friends & family	Emotional cost: they don’t want to hurt your feelings or spark conflict at Thanksgiving.	They observe long-term patterns (communication quirks, blind spots) you won’t see yourself.
Students, community members, or anyone who benefits from you	Gratitude or dependence can create a “gratitude tax”: criticizing might feel ungrateful.	They’re end-users of your teaching, content, or leadership and can flag issues you’d never notice.
Online followers / customers	Public criticism risks backlash from your fan base or other customers; private DM still isn’t fully safe.	They experience the output of your work directly and can report real-world impact.

Why anonymity flips the cost-benefit equation

1. Removes fear of retaliation or judgment – multiple workplace studies show people give more honest, detailed feedback when names are hidden (Lattice).
2. Reduces social-desirability bias – even peers who could speak up often soften the message to avoid awkwardness; anonymity lets them transmit the unfiltered signal (feedbackfruits.com).
3. Encourages participation from the quiet majority – in surveys, the people who rarely speak in meetings are the ones most likely to use an anonymous channel (business.com).
4. Makes it easy to “taste while cooking” – by lowering friction (a short form link), you collect a continuous trickle of observations instead of sporadic performance-review dumps.

Handling quality concerns

• Noise & trolls: yes, you’ll get “nonsense or nice,” as Hall says. Filter for patterns—three independent comments on the same issue beat one exquisitely argued email.
• Qualification of the critic: ask for context in the form (e.g., “How do you know my work?”). That lets you weigh comments from close collaborators more heavily without revealing their identity.
• Follow-up: aggregate feedback into themes you can act on—then tell your circle what you changed. When people see results, the quality of submissions improves.

Take-aways

• The very people whose opinions you should care about often have the most to lose socially by being frank; anonymity lowers that cost.
• Anonymous feedback isn’t a replacement for direct conversation, but a complementary “early-warning radar.” Use it to spot issues, then decide which ones merit a live discussion.
• Treat every anonymous comment as a data point, not a verdict—look for converging signals across many submissions.

Framed this way, anonymity isn’t about empowering random strangers; it’s about liberating credible insiders to help you “taste the dish” before you serve it.

Note: The source for what Cate Hall said is here.

Examples of monoids

Examples of monoids

A monoid is an algebraic structure with a single associative binary operation and an identity element. Formally, a monoid is a set $M$ with a binary operation $\cdot$ such that:

1. Closure: For all $a,b\in M$, $a\cdot b\in M$.
2. Associativity: For all $a,b,c\in M$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity: There exists an element $e\in M$ such that for all $a\in M$, $e\cdot a=a\cdot e=a$.

Here are some common examples of monoids:

---

1. Natural Numbers with Addition

• Set: ${\mathbb{N}}_0=0,1,2,\dots$
• Operation: Addition $(+)$
• Identity: 0

---

2. Strings with Concatenation

• Set: All strings over an alphabet ${\mathrm{\Sigma }}^{\ast }$
• Operation: String concatenation
• Identity: Empty string $\epsilon$

---

3. Integers with Multiplication

• Set: $\mathbb{Z}$
• Operation: Multiplication $(\cdot )$
• Identity: 1

---

4. Boolean Values with AND or OR

• Set: $\text{true},\text{false}$

  • With AND: Identity is true
  • With OR: Identity is false

---

5. Matrices with Matrix Multiplication

• Set: All $n\times n$ matrices over a field
• Operation: Matrix multiplication
• Identity: Identity matrix $I_n$

---

6. Functions from a Set to Itself (Endofunctions)

• Set: All functions $f:X\to X$
• Operation: Function composition
• Identity: Identity function ${\text{id}}_X$

---

7. Lists under Concatenation

• Set: Lists of elements (e.g., integers)
• Operation: Concatenation of lists
• Identity: Empty list []

---

Let me know if you want examples from programming (e.g., Haskell, Python), category theory, or more abstract algebra.

Allthetropes Origin Story

I assume allthetropes.org was created due to some drama with tvtropes.org? Do you have the scoop?

Yes — your assumption is basically right. All The Tropes (ALLTHETROPES.org) was born as a direct response to what many users saw as overreaching changes at TV Tropes.

---

⚠️ What Happened?

1. The Google Incidents & Content Removal

• In 2010 and again in 2012, Google flagged TV Tropes for hosting adult or sensitive tropes (like rape- or sexism-related content).
• TV Tropes responded by restricting or deleting entries automatically to comply with AdSense policies — leading to visible censorship and user frustration (Wikipedia, Wikipedia).

2. License & Contributor Policy Changes

• TV Tropes shifted its content license in July 2012, switching to the CC BY‑NC‑SA model.
• Later in 2013–2015, the site attempted to require contributors to grant exclusive ownership of their edits — a policy that was eventually revoked.
• This sparked clear discontent among volunteer editors (Wikipedia).

3. Forking with the Original License

• In response, a group of contributors scraped the entire TV Tropes database (back when it was under the more permissive CC BY‑SA license) and launched All The Tropes around mid‑2012 to mid‑2013.
• This fork was meant to preserve the original community content and to continue using the license under terms they felt were more equitable (Wikipedia).

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🧲 Community Voices

On Reddit, longtime fans summarize the split succinctly:

“All the tropes was formed by a group that broke from TV Tropes after the second google incident led to TV Tropes scrubbing pretty much all …” (Reddit)

“Don’t feel bad, there are several ‘no censorship’ forks out there. … A wikia, a .org and a spinoff of that org…” — referring to All The Tropes, Tropedia (the Wikia fork), and others (Reddit)

---

📌 Quick Timeline

Year	TV Tropes Event	Fork Response
~2010–2012	Google's AdSense crackdown → content removal	Censorship frustrations grow
July 2012	License moves to CC BY‑NC‑SA (non-commercial)	Raises legal and freedom concerns
≈2013	Contributors asked for exclusive rights to contributions	Many disapprove and look to fork
2013–2015	ALLTHETROPES launched as CC BY‑SA clone of TV Tropes	Community re-establishes a parallel wiki

---

✅ So, Was It Drama?

It was a blend of legal/copyright and policy conflicts, not necessarily personal drama. Contributors felt that TV Tropes’ editorial changes and licensing moves undermined the open spirit of the original wiki. All The Tropes was their way of preserving the prior ethos — no censorship, community-centered, and fully compatible with CC BY‑SA.

If you're interested in differences in UI, moderation style, or examples of tropes that were removed at TV Tropes but preserved on All The Tropes, I can dig into that too.

Mohammedan horses

OK so we know the names of various persons associated with the prophet Muhammad such as wives, followers, etc; do we know of any named horses?

Yes, several horses associated with the Prophet Muhammad ﷺ are mentioned by name in Islamic historical sources. These names appear in hadith collections, sīrah (biographies), and traditional literature. Here are the most well-known horses attributed to the Prophet:

1. Al-Burāq

• Not a horse in the conventional sense, but a heavenly steed.
• Used during the Isrāʾ and Miʿrāj (Night Journey and Ascension).
• Described as a white beast, larger than a donkey but smaller than a mule, with wings.
• Though not a horse per se, it's the most famous riding animal associated with the Prophet.

2. Sakb (سَكْب)

• Possibly the most frequently mentioned actual horse of the Prophet.
• Described as fast and agile, used in various battles.

3. Murtajiz (مُرْتَجِز)

• Named for the sound it made — “rajz” refers to a rhythmic sound or neigh.
• Reported to be a gift from al-Muqawqis, the ruler of Egypt.

4. Lahif (لاحق)

• A lesser-known horse, meaning “swift” or “overtaking.”

5. Lizaz (لِزَاز)

• Said to have been a gift from the Roman Emperor.

6. Ward (وَرْد)

• Meaning “rose” or “flower,” though possibly referring to its color.

7. Ya’fur (يعفور)

• Often mistakenly referred to as a donkey, not a horse. A famous story involves the Prophet having a donkey by this name that spoke to him.

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A Note on Authenticity:

The historical reliability of some of these names varies. Sakb and Murtajiz are more frequently referenced in hadith and biographies with higher authenticity. Others like Lizaz and Ward appear in later literature and may have weaker chains of transmission.

Would you like a breakdown of which sources mention each horse?

Yes please

Here are the most frequently mentioned named horses (and other riding animals) associated with the Prophet Muḥammad ﷺ—along with what our sources say about them:

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🐎 Named Horses of the Prophet

1. al‑Sakb (السكب)

• One of his earliest and most famous horses, meaning “The Flowing One” or “Swift,” named for its speed (Scribd, TikTok).
• It was reportedly the first horse he rode after arriving in Medina and he used it during the Battle of Uhud (Scribd).

2. al‑Murtajiz (مرتجز)

• Translating as “The Neigher,” named after its distinctive neigh (TikTok).
• Gifted by al‑Muqawqis, the Byzantine governor of Egypt (Scribd).

3. al‑Lizāz (Lizaz / Lazaz)

• Often listed among the presents from al‑Muqawqis (Scribd).

4. al‑Luhayf (also Luhaif / Laḥīf)

• Meaning “the one that drags” or “swift,” so named for its gait or carrying ability (Facebook).
• Also noted among gifts conveyed by supporters such as Farwah ibn ‘Amr al‑Judhami (Scribd).

5. al‑Ward (الورد)

• Meaning “rose,” this horse was reportedly given by Tamīm al‑Dārī but later passed on to ʿUmar ibn al‑Khaṭṭāb, who rode it in God’s cause (Scribd).

6. al‑Zārib (or al‑Zarīb)

• Mentioned in lists alongside Lizāz and Luhaif as among horses offered to the Prophet by companions or benefactors (Scribd).

7. al‑Yaʽsub (الْيَعسُوب)

• Some sources indicate it may have belonged to him, though details are scarce (Scribd).

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🐪 Other Named Mounts

• Yāʽfūr (يَعفُور): A donkey, not a horse, famously gifted by al‑Muqawqis. Islamic tradition includes miraculous stories about it speaking to the Prophet—though hadith scholars generally regard these as weak or inauthentic (Wikipedia).
• Duldul: A mule gifted alongside Yāʽfūr. Though not a horse, it is often mentioned among his mounts (Scribd).

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🧾 Summary Table

Name	Animal Type	Meaning / Notes	Source Notes
al‑Sakb	Horse	“Swift, like running water”	Frequently first horse in Medina, Uhud ride (Scribd)
al‑Murtajiz	Horse	“The Neigher”	Gift from al‑Muqawqis (Scribd, TikTok)
al‑Lizāz	Horse	Gifted to him	Also from al‑Muqawqis (Scribd)
al‑Luhayf	Horse	"Swift / dragging"	Carried him; gift from companions (Scribd, Facebook)
al‑Ward	Horse	“Rose”	Gifted then passed to ʿUmar (Scribd)
al‑Zārib	Horse	—	Mentioned among gifts (Scribd)
al‑Yaʽsub	Horse?	Possibly another mount; little detail	Rarely attested (Scribd)
Yāʽfūr	Donkey	Speaking donkey; gift from al‑Muqawqis	Treated separately (Wikipedia)
Duldul	Mule	Well-known mule; gift from same source	Often counted among his beasts (Scribd, Wikipedia)

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📜 On Authenticity & Source Quality

• The names al‑Sakb and al‑Murtajiz are more reliably attested in early sīra (biographies) and hadith-related literature.
• Others like Lizāz, al‑Ward, and al‑Yaʽsub are mentioned in later or less detailed narrations, often in compilations discussing his gifts and belongings.
• The legendary donkey Yāʽfūr and mule Duldul appear prominently in folklore and literary traditions, but scholars often consider their miraculous aspects weak or symbolic.

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✅ Final Thoughts

Yes—Islamic historical sources do record several named horses belonging to the Prophet ﷺ, with the most commonly cited being al‑Sakb and al‑Murtajiz. Several others appear in traditional companion lists of gifts and mounts. If you'd like, I can dig deeper into the original biographies (like Ibn Hishām, al‑Ṭabarī, or Ibn Isḥāq) or the isnād chains for a particular horse. Let me know!