On Group-Like Magmoids

Abstract: A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative multiplication, identities and inverses can be generalized when the total binary operation is replaced by a partial binary operation. Poloids, groupoids, skew-poloids, skew-groupoids, prepoloids, pregroupoids, skew-prepoloids and skew-pregroupoids are then defined in terms of generalized associativity, generalized identities and generalized inverses. Some basic results about these magmoids are derived, and connections between poloid-like and prepoloid-like magmoids, in particular semigroups, are described. Notably, analogues of the Ehresmann-Schein-Nampooribad theorem are proved.