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Groups of complexes of a representable lattice-ordered group

Published online by Cambridge University Press:  18 May 2009

R. D. Byrd ,
J. T. Lloyd  and
J. W. Stepp
R. D. Byrd
Affiliation:
University of Houston
J. T. Lloyd
Affiliation:
University of Houston
J. W. Stepp
Affiliation:
University of Houston
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In 1954 N. Kimura proved that each idempotent in a semigroup is contained in a unique maximal subgroup of the semigroup and that distinct maximal subgroups are disjoint [13] (or see [6, pp. 21–23]). This generalized earlier results of Schwarz [14] and Wallace [15]. These maximal subgroups are important in the study of semigroups. If G is a group, then the collection S(G) of nonempty complexes of G is a semigroup and it is natural to inquire what properties of G are inherited by the maximal subgroups of S(G). There seems to be very little literature devoted to this subject. In [5, Theorem 2], with certain hypotheses placed on an idempotent, it was shown that if G is a lattice-ordered group (“1-group”) then a maximal subgroup of S(G) containing an idempotent satisfying these conditions admits a natural lattice-order. The main result of this note (Theorem 1) is that if Gis a representable 1-group and E is a normal idempotent of S(G) and a dual ideal of the lattice G, then the maximal subgroup of S(G) containing E admits a representable lattice-order.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 19 , Issue 2 , July 1978 , pp. 135 - 139
Copyright
Copyright © Glasgow Mathematical Journal Trust 1978

References

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