Several types of groupoids induced by two-variable functions

Title
Several types of groupoids induced by two-variable functions
Author
김희식
Keywords
Groupoid; (Twisted) semigroup; Linear groupoid over a field; nth power property; Homomorphism
Issue Date
2016-10
Publisher
SPRINGER INTERNATIONAL PUBLISHING AG
Citation
SPRINGERPLUS, v. 5, Page. 1715-1725
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 phi : X-2 -˃ X-2 is a function phi (a, b) = (u, v), then (X, *) is a left-twisted semigroup with respect to phi if for all a, b, c is an element of 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 is an element of 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 = lambda x + mu y, with lambda, mu is an element of 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.
URI
https://springerplus.springeropen.com/articles/10.1186/s40064-016-3411-yhttp://repository.hanyang.ac.kr/handle/20.500.11754/99010
ISSN
2193-1801
DOI
10.1186/s40064-016-3411-y
Appears in Collections:
COLLEGE OF NATURAL SCIENCES[S](자연과학대학) > MATHEMATICS(수학과) > Articles
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