Quasigroup

Quasigroups differ from groups mainly in that the associative and identity element properties are optional.

A quasigroup (Q, ∗) is a non-empty set Q with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy closure property), obeying the Latin square property.

With regard to the Cayley table, the first equation (left division) means that the b entry in the a row is in the x column while the second equation (right division) means that the b entry in the a column is in the y row.

Algebraic structures that satisfy axioms that are given solely by identities are called a variety.

A quasigroup (Q, ∗, \, /) is a type (2, 2, 2) algebra (i.e., equipped with three binary operations) that satisfy the identities:[b]

In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.

A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed idempotent".

This is equivalent to any one of the following single Moufang identities holding for all x, y, z: According to Jonathan D. H. Smith, "loops" were named after the Chicago Loop, as their originators were studying quasigroups in Chicago at the time.

[10] A narrower class is a totally symmetric quasigroup (sometimes abbreviated TS-quasigroup) in which all conjugates coincide as one operation: x ∗ y = x / y = x \ y.

Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).

A quasigroup (Q, ∗) is called weakly totally anti-symmetric if for all c, x, y ∈ Q, the following implication holds.

[11] A quasigroup (Q, ∗) is called totally anti-symmetric if, in addition, for all x, y ∈ Q, the following implication holds:[11] This property is required, for example, in the Damm algorithm.

The inverse mappings are left and right division, that is, In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on universal algebra) are where id denotes the identity mapping on Q.

The multiplication table of a finite quasigroup is a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.

Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements.

For a countably infinite quasigroup Q, it is possible to imagine an infinite array in which every row and every column corresponds to some element q of Q, and where the element a ∗ b is in the row corresponding to a and the column responding to b.

In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.

For an uncountably infinite quasigroup, such as the group of non-zero real numbers under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in a sequence.

Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that for all x, y in Q.

An isotopy is a homotopy for which each of the three maps (α, β, γ) is a bijection.

Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation.

That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of ∗.

Polyadic or multiary means n-ary for some nonnegative integer n. A 0-ary, or nullary, quasigroup is just a constant element of Q.

Finite irreducible n-ary quasigroups exist for all n > 2; see Akivis & Goldberg (2001) for details.