Christopher Hollings

Theory of Generalised Heaps and Generalised Groups

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.

Investigating a claim for Russian priority in the abstract definition of a ring

In this article, I scutinize an assertion that the Russian-Ukrainian mathematician S O Shatunovskii (1859–1929) should be credited with the first modern definition of a ring. Shatunovskii’s claim is compared with that of Abraham Fraenkel, who defined a notion very close to the current concept of a ring in a paper of 1914.

A Ternary Algebraic Operation in the Theory of Coordinate Structures

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 (φ3φ2φ1) = φ3φ2−1φ1, 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.

The early mathematical education of Ada Lovelace

Ada, Countess of Lovelace, is remembered for a paper published in 1843, which translated and considerably extended an article about the unbuilt Analytical Engine, a general-purpose computer designed by the mathematician and inventor Charles Babbage. Her substantial appendices, nearly twice the length of the original work, contain an account of the principles of the machine, along with a table often described as ‘the first computer program’. In this paper we look at Lovelaces education before 1840, which encompassed older traditions of practical geometry; newer textbooks influenced by continental approaches; wide reading; and a fascination with machinery. We also challenge judgements by Dorothy Stein and by Doron Swade of Lovelaces mathematical knowledge and skills before 1840, which have impacted later scholarly and popular discourse.

‘Nobody could possibly misunderstand what a group is’: a study in early twentieth-century group axiomatics

In the early years of the twentieth century, the so-called ‘postulate analysis’—the study of systems of axioms for mathematical objects for their own sake—was regarded by some as a vital part of the efforts to understand those objects. I consider the place of postulate analysis within early twentieth-century mathematics by focusing on the example of a group: I outline the axiomatic studies to which groups were subjected at this time and consider the changing attitudes towards such investigations.

A tale of mathematical myth-making: E T Bell and the ‘arithmetization of algebra’

Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.

Mathematics Emerging: from Colorado to Oxford

Abstract History of mathematics courses based on original source materials are becoming increasingly common. However, are they more suitable for particular types of students? Here we compare two such upper-level courses, with a similar structure and using the same textbook, taken by liberal arts students at Colorado College and mathematics specialists at Oxford University.

Notes on the Translations

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

Viktor Vladimirovich Wagner (1908–1981)

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.

Generalised Heaps as Affine Structures

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.