a set with an n-ary operation, with , such that the equation has a unique solution for any one variable if all the other n variables are specified arbitrarily. 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 binary, or 2-ary, quasigroup is an ordinary quasigroup. An example of a multiary quasigroup is an iterated group operation, ; it is not necessary to use parentheses to specify the order of operations because the group is associative. 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. There exist multiary 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 and . Finite irreducible n-ary quasigroups exist for all ; see Akivis and Goldberg (2001) for details. An n-ary quasigroup with an n-ary version of associativity is called an n-ary group. Right- and left-quasigroups A right-quasigroup is a type (2,2) algebra satisfying both identities: y = (y / x) ∗ x; y = (y ∗ x) / x. Similarly, a left-quasigroup is a type (2,2) algebra satisfying both identities: y = x ∗ (x \ y); y = x \ (x ∗ y). Number of small quasigroups and loops The number of isomorphism classes of small quasigroups and loops is given here: See also Division ring – a ring in which every non-zero element has a multiplicative inverse Semigroup – an algebraic structure consisting of a set together with an associative binary operation Monoid – a semigroup with an identity element Planar ternary ring – has an additive and multiplicative loop structure Problems in loop theory and quasigroup theory Mathematics of Sudoku Notes References External links quasigroups Non-associative algebra Group theory Latin squares