Associative quasigroups: They are groups

Explain associative quasigroups

Sure. Here’s the picture in full:

1. Definitions

• A quasigroup $(Q,*)$ is a set with a binary operation $*$ such that for every $a,b\in Q$ the equations $a*x=b$ and $y*a=b$ 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 $e*a = a*e = a$ for all $a$. (Wikipedia)

• An associative quasigroup is a quasigroup whose operation is associative: $(a*b)*c = a*(b*c)$ 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 $(Q,*)$ 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 $(Q,*)$ 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 $a*e = a$. For an arbitrary $b\in Q$, pick $y$ with $y*a = b$; then

  $$ b*e = (y*a)*e = y*(a*e) = y*a = b $$

  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 $a*x = e$ and $y*a = e$. 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 $a*x=b$ and $y*a=b$). (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

  $$ v=a_1e_1+\dots+a_ne_n, $$

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

• Linear independence: If $\lambda_1e_1+\dots+\lambda_ne_n=0$ 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 $n$.

$$ \boxed{\dim_{\mathbb{F}_q}\bigl(\mathbb{F}_q^{\,n}\bigr)=n.} $$

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).
• $a_0, a_1, \ldots, a_n$ 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 $f(k) \neq 0$.
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 $a\, R\, b$, then $b\, R\, a$

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 $a\, R\, b$ and $b\, R\, a$, 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:

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1. Natural Numbers with Addition

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

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2. Strings with Concatenation

• Set: All strings over an alphabet $\Sigma^*$
• Operation: String concatenation
• Identity: Empty string $\varepsilon$

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3. Integers with Multiplication

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

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4. Boolean Values with AND or OR

• Set: $\{\text{true}, \text{false}\}$

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

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5. Matrices with Matrix Multiplication

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

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6. Functions from a Set to Itself (Endofunctions)

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

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7. Lists under Concatenation

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

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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.