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Anti-uniform semilattices

Published online by Cambridge University Press:  17 April 2009

J. M. Howie  and
B. M. Schein
J. M. Howie
Affiliation:
University of Stirling, Stirling, Scotland, and University of Saratov, Saratov, USSR.
B. M. Schein
Affiliation:
University of Stirling, Stirling, Scotland, and University of Saratov, Saratov, USSR.
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Abstract

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An inverse semigroup which is a union of groups is called Cliffordian. A semilattice E is called universally Cliffordian if every inverse semigroup having E as semilattice of idempotents is Cliffordian. It is shown that E is universally Cliffordian if and only if it is anti-uniform, that is, if and only if no two distinct principal ideals of E are isomorphic.

A semilattice E satisfying the minimum condition is anti-uniform if and only if it is a well-ordered chain. Examples are given of anti-uniform semilattices of more complicated types.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 2 , August 1969 , pp. 263 - 268
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Chang, C.C. and Ehrenfeucht, A., “A characterisation of abelian groups of automorphisms of a simply ordering relation”, Fund. Math. 51 (1962), 141–147.CrossRefGoogle Scholar
[2]Clifford, A.H., “Semigroups admitting relative inverses”, Ann. of Math. (2) 42 (1941), 1037–1049.Google Scholar
[3]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, vols. 1 and 2, (Amer. Math. Soc. Mathematical Surveys 7, Providence, 1961 and 1967).Google Scholar
[4]Howie, J.M., “The maximum idempotent-separating congruence on an inverse semigroup”, Proc. Edinburgh Math. Soc. (2) 14 (1964), 71–79.CrossRefGoogle Scholar
[5]Morel, Anne C., “On the arithmetic of order types”, Trans. Amer. Math. Soc. 92 (1959), 48–71.CrossRefGoogle Scholar
[6]Munn, W.D., “Uniform semilattices and bisimple inverse semigroups”, Quart. J. Math. Oxford. (2) 17 (1966), 151–159.CrossRefGoogle Scholar
[7]Schein, B.M., “Generalized groups with the well-ordered set of idempotents”, Mat.-Fyz. Časopis Sloven. Akad. Vied. 14 (1964), 259–262.Google Scholar