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Several types of groupoids induced by two-variable functions.

P J Allen1, Hee Sik Kim2, J Neggers1

  • 1Department of Mathematics, University of Alabama, 35487-0350 Tuscaloosa, AL USA.

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|October 26, 2016
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Summary

This paper explores twisted semigroups, a novel concept extending groupoids and semigroups with modified associative laws. We investigate their properties and introduce specific types like left-twisted and right-twisted semigroups.

Keywords:
(Twisted) semigroupGroupoidHomomorphismLinear groupoid over a field[Formula: see text] power property

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Area of Science:

  • Algebraic Structures
  • Abstract Algebra
  • Mathematical Structures

Background:

  • Semigroups are fundamental algebraic structures with associative binary operations.
  • Groupoids are generalized algebraic structures that may not satisfy associativity.
  • The study of algebraic structures with modified laws is an active area of research.

Purpose of the Study:

  • To introduce and define various types of twisted semigroups derived from groupoids.
  • To explore the properties of these twisted semigroups, including their duals and homomorphisms.
  • To investigate groupoids defined over a field and their behavior as twisted semigroups.

Main Methods:

  • Definition of left-twisted, right-twisted, and middle-twisted semigroups based on a groupoid and a function.
  • Examination of dual structures for twisted semigroups.
  • Analysis of groupoids defined over a field using specific formulas and structure constants.
  • Investigation of homomorphisms between twisted semigroups.

Main Results:

  • Formal definitions and examples of different types of twisted semigroups are provided.
  • Properties of groupoids defined over a field are discussed in the context of twisted semigroups.
  • A key result shows that simultaneous left-twistedness and right-twistedness implies the structure is a semigroup.

Conclusions:

  • Twisted semigroups offer a new perspective on algebraic structures by relaxing the associative law.
  • Groupoids defined over fields exhibit interesting properties when viewed as twisted semigroups.
  • The interplay between different types of twistedness can lead to the recovery of standard semigroup properties.