Biunit pairs in semiheaps and associated semigroups

TL;DR

This paper introduces biunit pairs in semiheaps, explores their connection to semigroups via switches, and defines diheaps, expanding the algebraic framework linking semiheaps and semigroups.

Contribution

It establishes a one-to-one correspondence between monoids with switches and semiheaps with biunit pairs, generalizing existing semiheap-semigroup relations.

Findings

Biunit pairs generalize units in semiheaps.

A correspondence between monoids with switches and semiheaps with biunit pairs is proven.

Diheaps are introduced and shown to be warp-equivalent but non-isomorphic to heaps.

Abstract

Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of unit. Families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a biunit pair. This generalizes a well-established result in semiheap theory that connects involuted semigroups and semiheaps with biunit elements. Furthermore, diheaps are introduced as semiheaps whose elements belong to biunit pairs and they are shown to be non-isomorphic but warp-equivalent to heaps.