title: Several types of groupoids induced by two-variable functions.
abstract: 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.
authors: Department of Mathematics, University of Alabama, 35487-0350 Tuscaloosa, AL USA.
authors: P J
authors: PJ
authors: Allen
authors: Research Institute for Natural Sciences, Department of Mathematics, Hanyang University, 04763 Seoul, Korea.
authors: Hee Sik
authors: HS
authors: Kim
authors: Department of Mathematics, University of Alabama, 35487-0350 Tuscaloosa, AL USA.
authors: J
authors: J
authors: Neggers
keywords: (Twisted) semigroup; Groupoid; Homomorphism; Linear groupoid over a field; [Formula: see text] power property