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Lattices of congruences on free finitely generated commutative semigroups and direct products of cyclic monoids

Published online by Cambridge University Press:  14 November 2011

D. C. Trueman
D. C. Trueman
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria, Australia2168
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Synopsis

Let S11× V1 be a direct product of a cyclic monoid S11with S1 not a group, and a semigroup V1 with an adjoined identity. We prove thatboth the lattice of congruences on (S11 × V1)/{(1,1)} and the lattice of congruences on (S11 × V1) are neither lower semimodular nor uppersemimodular. We then prove that the lattice of congruences on a free finitely generated commutative semigroup with more than one generator is neither lower semimodular nor upper semimodular.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 175 - 179
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Clifford, A. H. and Preston, G. B.. The algebraic theory of semigroups. Providence, R. I.: Amer. Math. Soc. Surveys 7/I and II (1961 and 1967).Google Scholar
2Trueman, D. C.. The lattice of congruences on direct products of cyclic semigroups and certain other semigroups. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 203–214.CrossRefGoogle Scholar