  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.

Rings and Algebras