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In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative.
There are two (equivalent) formal definitions of quasigroup in common use, one of which defines a quasigroup to be a set with one binary operation, and the other which defines a quasigroup to be a set with three binary operations. We begin with the former definition, which is easier to follow.
A quasigroup (Q, *) is a set Q with a binary operation * : Q × Q → Q (that is, it is a magma or groupoid), such that for each a and b in Q, there exist unique elements x and y in Q such that
In the universal algebra approach, a quasigroup (Q, *, \, /) is defined to be a set Q with three binary operations (*, \, /) satisfying the following identities:
If (Q, *) is a quasigroup according to the first definition, then with its induced left and right division operations, (Q, *, \, /) is a quasigroup in the universal algebra sense. Conversely, if (Q, *, \, /) is a quasigroup in the second sense, then by simply "forgetting" the left and right division operations, we see that (Q, *) is a quasigroup in the first sense.
A loop is a quasigroup with an identity element e:
<small>In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.</small>
Quasigroups have the cancellation property: if ab = ac, then b = c. This is because x = b is certainly a solution of the equation ab = ax, and the solution is required to be unique. Similarly, if ba = ca, then b = c.
The definition of a quasigroup Q says that the left and right multiplication operators defined by are bijections from Q to itself. A magma Q is a quasigroup precisely when these operators are bijective. The inverse maps are given in terms of left and right division by In this notation the quasigroup identities are
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 every permutation of the elements, see small Latin squares and quasigroups.
Every loop has a unique left and right inverse given by
A loop is said to have (two-sided) inverses if for all x. In this case the inverse element is usually denoted by .
There are some stronger notions of inverses in loops which are often useful:
A loop has the inverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop which satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
Any loop which satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
A quasigroup or loop homomorphism is a map f : Q → P between two quasigroups such that f(xy) = f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
Let Q and P be quasigroups. 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.
An isotopy is a homotopy for which each of the three maps (α, β, γ) is a bijection. Two quasigroups are isotopic if there is an isotopy between them. 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. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup.
Each quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup which is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by (x+y)/2 is isotopic to the additive group R, but is not itself a group.
An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: Q<sup>n</sup> → Q, such that the equation f(x<sub>1</sub>,...,x<sub>n</sub>) = y has a unique solution for any one variable if all the other n variables are specified arbitrarily. Polyadic or multary means n-ary for some nonnegative integer n.
A 0-ary quasigroup is just a constant element of Q. A 1-ary quasigroup is a bijection of Q to itself.
An example of a multary quasigroup is an iterated group operation, y = x<sub>1</sub> · x<sub>2</sub> · ··· · x<sub>n</sub>; then it is not necessary to use parentheses because the group is associative. One can also carry out a sequence of the same or different group or quasigroup operations, if the order of operations is specified.
There exist multary quasigroups that cannot be represented in any of these ways. An n-ary quasigroup is irreducible if its operation cannot be factored into the composition of two operations in the following way: where 1 ≤ i < j ≤ n and (i, j) ≠ (1, n). Finite irreducible n-ary quasigroups exist for all n > 2; see Akivis and Goldberg (2001) for details.