Biunit pairs in semiheaps and associated semigroups

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.