# Several Types of Groupoids Induced by Two-Variable Functions

 dc.contributor.author Allen, P.J. dc.contributor.author Kim, Hee Sik dc.contributor.author Neggers, J. dc.date.accessioned 2021-07-13T13:05:51Z dc.date.available 2021-07-13T13:05:51Z dc.date.issued 2016 dc.identifier.citation Allen, P., Kim, H., Neggers, J. (2016): Several Types of Groupoids Induced by Two-Variable Functions. SpringerPlus. Volume 5. en_US dc.identifier.uri http://ir.ua.edu/handle/123456789/7946 dc.description.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. en_US dc.description.uri https://doi.org/10.1186/s40064-016-3411-y dc.format.mimetype application/pdf dc.language English en_US dc.rights.uri http://creativecommons.org/licenses/by/4.0/ dc.subject Groupoid en_US dc.subject (Twisted) semigroup en_US dc.subject Linear groupoid over a field en_US dc.subject nth power property en_US dc.subject Homomorphism dc.title Several Types of Groupoids Induced by Two-Variable Functions en_US dc.type text
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