Quasigroup in a sentence

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. A quasigroup homomorphism is just a homotopy for which the three maps are equal.

One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.

Polyadic or multiary means n-ary for some nonnegative integer n. A 0-ary, or nullary, quasigroup is just a constant element of Q. A 1-ary, or unary, quasigroup is a bijection of Q to itself.

A magma Q is a quasigroup precisely when all these operators, for every x in Q, are bijective.

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

Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system.

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.

Every quasigroup is isotopic to a loop.

However, a quasigroup which is isotopic to a group need not be a group.

In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ. An autotopy is an isotopy from a quasigroup to itself.

Isostrophe (Paratopy) If the set Q has two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be isostrophic to each other.

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

The empty set equipped with the empty binary operation satisfies this definition of a quasigroup.

The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup.