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A coordinatisation of implicative (semi-)lattices by Baer semigroups*

Published online by Cambridge University Press:  14 November 2011

T. S. Blyth
T. S. Blyth
Affiliation:
Mathematical Institute, University of St Andrews
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Synopsis

We show that if L is a bounded (semi-)lattice and f is a range-closed idempotent residuated mapping on L then f is multiplicative if and only if it is decreasing. We then use this to prove that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a weakly multiplicative (right) Baer semigroup.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 82 , Issue 1-2 , 1978 , pp. 111 - 113
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Blyth, T. S. and Janowitz, M. F.. Residuation theory (Oxford: Pergamon Press. 1972).Google Scholar
2Janowitz, M. F.. Decreasing Baer semigroups. Glasgow Math. J. 10 (1969) 46–41.CrossRefGoogle Scholar
3Johnson, C. S.. Semigroups coordinatising posets and semilattices. J. London Math. Soc. 4 (1971) 277–283.CrossRefGoogle Scholar