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Modular inverse semigroups

Part of: Semigroups Algebraic structures

Published online by Cambridge University Press:  09 April 2009

Katherine G. Johnston ,
Peter R. Jones  and
T. E. Hall
Katherine G. Johnston
Affiliation:
Department of Mathematics, College of Charleston, Charleston, South Carolina 29424, 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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Abstract

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An inverse semigroup S is said to be modular if its lattice 𝓛𝓕 (S) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each 𝓓-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author a simple modular inverse semigroup is “almost” distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

MSC classification

Secondary: 20M10: General structure theory 08A30: Subalgebras, congruence relations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 1 , August 1987 , pp. 47 - 63
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Birkhoff, G., Lattice theory (3rd ed., Amer. Math. Soc. Colloq. Publ., Providence, R. I., 1967).Google Scholar
[2]Clifford, A. H., ‘A class of d-simple semigroups’, Amer. J. Math. 15 (1953), 541–556.Google Scholar
[3]Fuchs, L., Infinite abelian groups, Vol. 1 (Academic Press, New York, 1970).Google Scholar
[4]Grätzer, G., General lattice theory (Academic Press, New York, 1978).CrossRefGoogle Scholar
[5]Howie, J. M., An introduction to semigroup theory (Academic Press, London, 1976).Google Scholar
[6]Jones, P. R., ‘Semimodular inverse semigroups’, J. London Math. Soc. 17 (1978), 446–456.CrossRefGoogle Scholar
[7]Jones, P. R., ‘Distributive inverse semigroups’, J. London Math. Soc. 17 (1978), 457–466.CrossRefGoogle Scholar
[8]Reilly, N. R., ‘Bisimple inverse semigroups’, Trans. Amer. Math. Soc. 132 (1968), 101–114.Google Scholar
[9]Reilly, N. R., ‘Congruences on a bisimple inverse semigroup in terms of RP-systems’, Proc. London Math. Soc. (3) 23 (1971), 99–127.CrossRefGoogle Scholar
[10]Petrich, M., Inverse semigroups (Wiley, New York, 1984).Google Scholar
[11]Suzuki, M., Structure of a group and the structure of its lattice of subgroups (Springer, Berlin, 1956).CrossRefGoogle Scholar