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Published in:
2017 | OriginalPaper | Chapter
7. Generalised Groups
Authors : Christopher D. Hollings, Mark V. Lawson
Published in: Wagner’s Theory of Generalised Heaps
Publisher: Springer International Publishing
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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.