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 (X, ∗) is a groupoid and if ϕ : X2 → X2 is a function ϕ(a, b) = (u, v), then (X, ∗) is a left-twisted semigroup with respect to ϕ if for all a, b, c ∈ X, a ∗ (b ∗ c) = (u ∗ v) ∗ c. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule (a ∗ b) ∗ c = u ∗ (v ∗ c) for all a, b, c ∈ X. 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 (X, +, ·) via a formula x ∗ y = ¬x + µy, with ¬,µ ∈ X, 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 (X, ∗) implies the fact that (X, ∗) is a semigroup.