A coalgebraic approach to quasigroup permutation representations

Abstract

The paper identifies the class of all permutation representations of a given finite quasigroup as a covariety of coalgebras. Each permutation representation decomposes as a sum of homomorphic images of homogeneous spaces. For a group, permutation representations in the present sense specialise to the classical concept. Burnside's Lemma, with a new proof, is extended from groups to quasigroups.

Permutation representations of left quasigroups

“…The abstract significance of the tensor product is given by Corollary 7.9 below. (Contrary to an erroneous claim in [19], it does not give a product in IFS Q . )…”

The Burnside algebra of a quasigroup

“…Nevertheless, the category Q is actually bicomplete (cf. Proposition 6.2(c) of [20]). Limits in the covariety Q are constructed by a procedure dual to that used for the construction of colimits in a (pre)variety of τ -algebras of a given type τ (cf.…”

“…Theorem 6.2 [20]. For a finite quasigroup Q, the Q-sets are precisely the sums of homomorphic images of homogeneous spaces.…”

“…Dual to the closure of a variety of algebras under homomorphic images, subalgebras and products, covarieties of coalgebras are closed under subcoalgebras, homomorphic images and sums. Thus general Q-sets over a quasigroup Q are defined according to [20] as members of the covariety Q of coalgebras generated by the homogeneous spaces over Q. In this way the troublesome products are avoided.…”

“…For a finite quasigroup Q, products of homogeneous spaces in the category of Q-IFS decompose into members of a potentially infinite set of isomorphism classes of indecomposable summands. To avoid such "wildness", quasigroup homogeneous spaces were construed in [20] as coalgebras. Dual to the closure of a variety of algebras under homomorphic images, subalgebras and products, covarieties of coalgebras are closed under subcoalgebras, homomorphic images and sums.…”