Several types of groupoids induced by two-variable functions.

In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if [Formula: see text] is a groupoid and if [Formula: see text] is a function [Formula: see text], then [Formula: see text] is a left-twisted semigroup with respect to [Formula: see text] if for all [Formula: see text], [Formula: see text]. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule [Formula: see text] for all [Formula: see text]. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field [Formula: see text] via a formula [Formula: see text], with [Formula: see text], fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of [Formula: see text] implies the fact that [Formula: see text] is a semigroup.