Theory of Generalised Heaps and Generalised Groups

Abstract

The purpose of this long paper was to develop the theories of generalised heaps and generalised groups in their mutual connections. To this end, Wagner began by introducing the new notion of a semiheap: a system with a ternary operation satisfying certain conditions. He explored some of the basic properties of semiheaps, as well as setting out elements of the theory of binary relations, the use of which was central to his approach. He next moved to the consideration of semigroups with involution, which turn out to have a natural connection with semiheaps, namely that any semiheap may be embedded in such a semigroup. Wagner then restricted his attention to a specific class of semigroups with involution: generalised groups (a.k.a. inverse semigroups), and the class of semiheaps with which they are closely associated: generalised heaps. He established elementary theories for these objects, and showed, for example, that any generalised heap may be embedded in a generalised group. These theories were then further expanded via the exploration of certain special binary relations in generalised heaps and generalised groups: the compatibility relation and the canonical order relation. The final section of the paper applies the previously developed notions to the context of binary relations and partial mappings and transformations: semiheaps and generalised heaps have a natural interpretation as abstractions of systems of binary relations or partial mappings between different sets, whilst semigroups and generalised groups apply in the case of partial transformations of a single set. It is proved that every generalised heap admits a representation by means of partial mappings, whilst every generalised group admits a representation via partial transformations.