SEMIGROUP DISTANCES OF FINITE GROUPOIDS

Abstract

The simplest cases of non-associative groupoids are presented by groupoids (so called SH-groupoids) having just one non-associative (ordered) triple of elements.  In this paper, only SH-groupoids having the simplest possible non-associative triple (a, a, a) are investigated.  For each positive integer n finite SH-groupoids E_n(·) generated by at most two elements are constructed and their semigroup distances are described.  It is proved that there are finite non-associative groupoids having their semigroup distance equal just to the arbitrary given positive integer n.