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On Axioms for Semi-Lattices

Published online by Cambridge University Press:  20 November 2018

R. Padmanabhan
R. Padmanabhan*
Affiliation:
University of Madras, Madurai and University of Panjab, Chandigarh, India
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By a semi-lattice we mean a system <L,.> where L is a set and. is a binary operation in L that is idempotent, commutative and associative. In a recent article [2] D.H. Potts considers the problem of reducing the number of axioms for a semi-lattice. His result was that the following two axioms viz. (1) xx=x, (2) (uv)((wx)(yz)) = ((uv)(xw))(zy) are sufficient to give a semi-lattice. But the second identity contains six elements instead of the original three. In the following we give a set of two simple identities with just three elements for a semi-lattice. This improves the above mentioned result.

Type
Notes and Problems
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 3 , August 1966 , pp. 357 - 358
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. McKinsey, J. C. C. and Diamond, A. H., Algebras and their subalgebras, Bull. Amer. Math. Soc. 53 (1947).Google Scholar
2. Potts, D.H., Axioms for semi-lattices, Canad. Math. Bull. 8 (1965) no. 4.Google Scholar