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Generalised Groups

In: Wagner’s Theory of Generalised Heaps

Author

Listed:
  • Christopher D. Hollings

    (University of Oxford, Mathematical Institute
    The Queen’s College)

  • Mark V. Lawson

    (Heriot-Watt University, Department of Mathematics)

Abstract

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.

Suggested Citation

  • Christopher D. Hollings & Mark V. Lawson, 2017. "Generalised Groups," Springer Books, in: Wagner’s Theory of Generalised Heaps, chapter 0, pages 43-47, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-63621-4_7
    DOI: 10.1007/978-3-319-63621-4_7
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