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The Boolean ring universal over a meet semilattice

Published online by Cambridge University Press:  09 April 2009

Elliot Evans
Elliot Evans
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, 37916, U.S.A.
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Abstract

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The Boolean ring B{M} universal over a meet semilattice M is examined. It is the vector space over the two element field Z2 with base M\{0}. The Z2 linear independence of a meet subsemilattice of a Boolean ring is characterized in order theoretic terms and some ramifications of this on B[M] are considered. The space (ℱpM) of proper filters of M is shown homeomorphic to the Stone space S(B[M]) of B[M] if M has no least element, with ℱP(M) ∪ {M} and S(B[M]) homeomorphic otherwise. The congruence lattice θ(M) of M is compared to the ideal lattice ℱ(B [M]) of B [M] with best results coming if M is a tree with zero when θ (M) = ℐ (B [M]).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 4 , June 1977 , pp. 402 - 415
Copyright
Copyright © Australian Mathematical Society 1977

References

Byrd, R. D., Mena, R. A. and Troy, L. A. (1975), ‘Generalized Boolean Lattices’, J. Austral. Math. Soc. 19, 225–237.Google Scholar
Clifford, A. H. and Preston, G. B. (1961). The Algebraic Theory of Semigroups, v. 1 (American Mathematical Society. Providence).Google Scholar
MacNeille, H. M. (1939), ‘The Extension of a distributive lattice to a Boolean ring’, Bull. Amer. Math. Soc. 45, 452–455.Google Scholar
Mostowski, A. and Tarski, A. (1939). ‘Boolesche Ringe mit Geordneter Basis’, Fund. Math. 32, 69–86.Google Scholar
Papert, D. (1964), ‘Congruence relations in Semilattices’, J. London Math. Soc. 39, 723–729.CrossRefGoogle Scholar
Schmidt, J. (1953). ‘Beitrage sur Filtertheorie II’, Math. Nachr. 10, 197–232.CrossRefGoogle Scholar