The Early Development of the Algebraic Theory of Semigroups

Hollings, Christopher

doi:10.1007/s00407-009-0044-3

Cited by 11 publications

(6 citation statements)

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“…Later sources, besides of those coming from the theory of groups and the theory of rings, include an abstraction of certain ideas arising in connection with topological or linear spaces. More about the history of algebraic theory of semigroups can be found in [17]. Within BISH, the history of constructive semigroups with an inequality began recently.…”

Section: Introductionmentioning

confidence: 99%

Semigroups with apartness

Crvenković

Mitrovic

Romano³

2013

Math. Log. Quart.

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Proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea used a constructive semigroup with inequality in . This motivated us to achieve a little progress in that direction. The starting point is the structure (S,=,≠,·) called a semigroup with apartness. Our primary objective is to prove isomorphism theorems for such constructive semigroups. In doing so our main ideas and notions come from .

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Section: Introductionmentioning

confidence: 99%

Semigroups with apartness

Crvenković

Mitrovic

Romano³

2013

Math. Log. Quart.

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“…He began by listing the axioms given by Speiser ( 1923 ) and asked (i) how can they be modified in order that a group still results, and (ii) what other structures can result? Prachar gave some references to instances of these other structures, for example, Brandt’s groupoids (in the case where we no longer insist that the multiplication be defined everywhere: see Hollings 2009a , §4, or Hollings 2014b , §6.2), and quasigroups (where associativity is weakened: see Pflugfelder 2000 ). Prachar’s motivation for studying this problem seems to have come ultimately from a 1935 group theory lecture given by Philipp Furtwängler and attended by Prachar’s doctoral supervisor Edmund Hlawka.…”

Section: A Huntington’s 1904 Postulatesmentioning

confidence: 99%

“… 59 Nowadays, the term ‘semigroup’ is of course used for a more general concept: a set with a (closed) associative binary operation. On the place of this work of de Séguier and Dickson in the development of the theory of semigroups, see Hollings ( 2009a , §3) or Hollings ( 2014b , §1.2). …”

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confidence: 99%

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

Hollings

2017

Arch. Hist. Exact Sci.

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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.

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“…Structure theorem is one of the first classical problems that arises in the study of any algebraic structure. Thus, it is no coincidence that the very first structure theorem in semigroups had close relations with the work of the very first 'proper' semigroup theorist Anton Suschkewitsch, as rightly recorded by Hollings [13]. In 1928, Suschkewitch [33] studied finite simple semigroups (i.e.…”

Section: Introductionmentioning

confidence: 99%

Cross-connection structure of concordant semigroups

Muhammed

Romeo

Nambooripad

2019

Int. J. Algebra Comput.

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Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. In this article, we describe the structure of locally inverse semigroups using Nambooripad's cross-connection theory. We characterise the categories involved as 'unambiguous categories' and provide a new structure theorem for locally inverse semigroups. Further, recall that there are two major structure theorems involving special classes of locally inverse semigroups: inverse semigroups (ESN Theorem via inductive groupoids) and completely 0-simple semigroups (via Rees matrix construction); they seem unrelated. In this article, we specialise our cross-connection description of locally inverse semigroups to these classes to exposit a unification of these different approaches to the structure theory of semigroups. In particular, we show that the structure theorem in inverse semigroups can be obtained using only one category, quite analogous to the ESN Theorem; in a completely 0-simple semigroup, we show that the cross-connection coincides with the structure matrix.

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The Early Development of the Algebraic Theory of Semigroups

Cited by 11 publications

References 105 publications

Semigroups with apartness

Semigroups with apartness

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

Cross-connection structure of concordant semigroups

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