Biunit pairs in semiheaps and associated semigroups

Bernard Rybołowicz*, Carlos Zapata-Carratalá

*Corresponding author for this work

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Abstract

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal’cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, 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 fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to heaps only when they are heaps themselves and explicit non-heap examples are constructed from abelian groups and hypermatrices.

Original languageEnglish
Pages (from-to)229-249
Number of pages21
JournalSemigroup Forum
Volume107
Early online date9 Aug 2023
DOIs
Publication statusPublished - Aug 2023

Keywords

  • Biunit
  • Involution
  • Semigroup
  • Semiheap
  • Ternary algebra

ASJC Scopus subject areas

  • Algebra and Number Theory

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