Wagner’s Theory of Generalised Heaps

We provide an overview of the articles by Wagner that are here translated, as well as indicating the contextual material that is provided.

We give a brief biographical sketch of the Russian mathematician Viktor Vladimirovich Wagner (1908–1981), together with an indication of the main themes of his work.

We briefly outline the relevant parts of the histories of the theories of binary relations, ternary operations, partial transformations and semigroups, as well as differential geometry, and describe how they came together in Wagner’s work.

We give an indication of some of the choices that have been made in the process of preparing the translations presented here.

In this short communication to the Academy of Sciences, Wagner took \(\mathfrak{M}(A \times B)\) to be the collection of all one-to-one partial mappings from a set A to a set B. A coordinate structure K on A is a subset of \(\mathfrak{M}(A \times B)\). A ternary operation can be defined in \(\mathfrak{M}(A \times B)\) by (φ ₃ φ ₂ φ ₁) = φ ₃ φ ₂ ⁻¹ φ ₁, where⁻¹ indicates the inverse of an injective partial mapping. Wagner’s main interest was in those coordinate structures that have closure properties with respect to this operation. The purpose of this paper seems to have been to introduce this formulation as a means of providing an abstract description of coordinate structures in differential geometry.

In this short communication to the Academy of Sciences, Wagner confined his attention to binary relations between the elements of a single set A, denoting the collection of all such relations by \(\mathfrak{P}(A \times A)\). He noted that the latter forms a semigroup under composition of binary relations; this semigroup is ordered by set inclusion and, moreover, has a natural involution, via which any binary relation is sent to its inverse. Wagner called a subset of \(\mathfrak{P}(A \times A)\) symmetric if it is closed under this involution; he identified the most important of the symmetric subsets of \(\mathfrak{P}(A \times A)\) as being \(\mathfrak{M}(A \times A)\), the collection of all one-to-one partial transformations of A. He proved that within \(\mathfrak{M}(A \times A)\) both the order relation and the involution may be expressed in terms of composition of transformations. Wagner went on to relate \(\mathfrak{M}(B \times B)\) to the group \(\mathfrak{G}(A \times A)\) of all bijections of A, for some A ⊃ B.

In this short communication to the Academy of Sciences, Wagner considered semigroups S in which for every element s there is a corresponding element s ^′ such that ss ^′ s = s and s ^′ ss ^′ = s ^′ . If, in addition, the idempotents commute in such a semigroup, then the element s ^′ is unique, is termed the generalised inverse of s and is denoted s ⁻¹. After proving basic results about generalised inverses, Wagner defined a generalised group (or inverse semigroup in modern terminology) to be a semigroup in which every element has a unique generalised inverse. He further defined a natural partial order in such a semigroup and derived certain of its basic properties. Turning to semigroups of partial transformations, Wagner noted that the semigroup of all one-to-one partial transformations of a set forms a generalised group, and, moreover, that any generalised group may be represented as one of these, a result now termed the Wagner–Preston Representation Theorem.

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.

We describe how Wagner’s notion of a generalised heap can be transformed into something much more akin to a generalised bitorsor with the help of Anders Kock’s concept of a pregroupoid. This directly generalises the classical relationship between heaps and bitorsors of groups. The generalised bitorsors that give rise to generalised heaps are, in fact, the exact arbiters of Morita equivalence between inverse semigroups. Thus generalised heaps are of much more than merely historical interest.