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.