Permutation representations of loops

A coalgebraic approach to quasigroup permutation representations

“…(Compare [17], Proposition 8.1, where this result was formulated for a loop Q. The proof given there applies to an arbitrary non-empty quasigroup Q.)…”

“…It is readily checked that the class of morphisms (3.4), for a fixed set Q, forms a concrete category IFS Q . For a group Q, it was shown in [17] that the category of finite Q-sets forms the full subcategory of IFS Q consisting of those objects for which the action map (3.1) is a monoid homomorphism. Moreover,…”

“…If Q is a loop with identity element e, then the regular homogeneous space may also be described as ({e} \ Q, Q). This definition was used in §7 of [17].) For a group Q, each homogeneous space (P \ Q, Q) is obtained as a homomorphic image of the regular permutation representation.…”

“…of permutations of the set X. In [17], an attempt was made to extend this algebraic approach to more general quasigroups. However, because of the demand for closure of the class of Q-sets under products, it turned out to be necessary to require Q to be a loop (a quasigroup with an identity element) 1 .…”

“…However, because of the demand for closure of the class of Q-sets under products, it turned out to be necessary to require Q to be a loop (a quasigroup with an identity element) 1 . An additional feature of the approach taken in [17] was the infinite number of non-isomorphic irreducible Q-sets that kept appearing in repeated direct products. Burnside's Lemma, which holds for quasigroup homogeneous spaces (Theorem 5.1 of [15]), no longer holds for these new irreducible Q-sets.…”