Quasigroups

Explain quasigroups

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

• $a * x = b$, and
• $y * a = b$

have unique solutions $x,y\in Q$. 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: $a\backslash b$ is the unique $x$ with $a * x = b$.
• Right division: $b / a$ is the unique $y$ with $y * a = b$.

These satisfy identities like

$$ a * (a\backslash b) = b,\quad (b / a) * a = b,\quad a\backslash(a * b) = b,\quad (b * a)/a = b. $$

Basic properties:

• Cancellation: From uniqueness of solutions, left and right cancellation hold: if $a * b = a * c$, then $b=c$; similarly $b * a = c * a$ implies $b=c$.
• Not necessarily associative: The operation need not satisfy $(a * b) * c = a * (b * c)$.
• 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 $e * a = a = a * e$ 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 $a * b = a - b \pmod 3$. Then for any $a,b$, $a * x = b$ 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 $(\mathbb{Z}_3, a * b = a - b)$ 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

$$ \begin{array}{c|ccc} * & 1 & 2 & 3\\\hline 1 & 2 & 3 & 1\\ 2 & 3 & 1 & 2\\ 3 & 1 & 2 & 3 \end{array} $$

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

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

  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” $\iff$ 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 $a * x = b$ and $y * a = b$ 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 $a^{L} * a = e$,
  • A right inverse $a^{R}$ such that $a * a^{R} = e$. These exist uniquely, but in a general loop $a^{L}$ and $a^{R}$ might be different. If they coincide and behave well (e.g., $(a^{-1})^{-1}=a$, $(a b)^{-1}=b^{-1}a^{-1}$), 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 $a^{-1} * a = a * a^{-1} = e$; associativity makes these inverses behave in the familiar algebraic way.

So: quasigroup division gives you the ability to “undo” $a *$ or $* a$ 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.