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The lattice of full regular subsemigroups of a regular semigroup

Published online by Cambridge University Press:  14 November 2011

Katherine G. Johnston  and
Peter R. Jones
Katherine G. Johnston
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, U.S.A.
Peter R. Jones
Affiliation:
Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisconsin 53233, U.S.A.
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Synopsis

Although the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S.

A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 3-4 , 1984 , pp. 203 - 214
Copyright
Copyright © Royal Society of Edinburgh 1984

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