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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

REFERENCES

1.Birkhoff, G., Lattice Theory (Amer. Math. Soc., 1967).Google Scholar
2.Bernau, S. J., Orthocompletions of lattice groups, Proc. London Math. Soc. 16 (1966), 107–130.CrossRefGoogle Scholar
3.Byrd, R. D., Complete distributivity in lattice-ordered groups, Pacific J. Math. 20 (1967), 423–432.CrossRefGoogle Scholar
4.Byrd, R. D., Lloyd, J. T. and Stepp, J. W., Groups of complexes of a group, J. Natur. Sci. and Math. 15 (1975), 83–87.Google Scholar
5.Byrd, R. D., Lloyd, J. T. and Stepp, J. W., Groups of complexes of a lattice-ordered group, Sym. Math. 21 (1977) 525–528.Google Scholar
6.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups Vol. I, Amer. Math. Soc. Mathematical Surveys 7 (Providence, R.I., 1961).CrossRefGoogle Scholar
7.Conrad, P. F., Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99 (1961), 212–240.CrossRefGoogle Scholar
8.Conrad, P. F., The relationship between the radical of a lattice-ordered group and complete distributivity, Pacific J. Math. 14 (1964), 493–499.CrossRefGoogle Scholar
9.Conrad, P. F., Archimedean extensions of lattice-ordered groups, J. Indian Math. Soc. 30 (1966), 131–160.Google Scholar
10.Conrad, P. F., Lattice-ordered groups, Lecture Notes (Tulane University, 1970).Google Scholar
11.Conrad, P. F., Epi-Archimedean groups, Czechoslovak Math. J. 24 (1974), 192–218.CrossRefGoogle Scholar
12.Fuchs, L., Partially ordered algebraic systems (Pergamon Press, 1963).Google Scholar
13.Kimura, N., Maximal subgroups of a semigroup, Kōdai Math. Sem. Rep. 1954 (1954), 85–88.Google Scholar
14.Schwarz, S., Zur Theorie der Halbgruppen (Slovakian, German summary), Sbornik prac Prirodovedekej Fakulty Slovenskej University v Bratislave No 6 (1943).Google Scholar
15.Wallace, A. D., A note on mobs II, An. Acad. Brasil Ci. 25 (1953), 335–336.Google Scholar