Translatability and translatable semigroups

TL;DR

This paper investigates the properties of k-translatable groupoids, establishing conditions under which they form semigroups and exploring their structural characteristics and construction methods.

Contribution

It provides new theoretical insights into k-translatable groupoids, including criteria for semigroup formation and methods for constructing non-left cancellative examples.

Findings

Necessary and sufficient conditions for semigroup formation

Characterization of semigroups as unions of cyclic groups

Construction methods for non-left cancellative k-translatable semigroups

Abstract

The concept of a k-translatable groupoid is explored in depth. Some properties of idempotent k-translatable groupoids, left cancellative k-translatable groupoids and left unitary k-translatable groupoids are proved. Necessary and sufficient conditions are found for a left cancellative k-translatable groupoid to be a semigroup. Any such semigroup is proved to be left unitary and a union of disjoint copies of cyclic groups of the same order. Methods of constructing k-translatable semigroups that are not left cancellative are given.