Monotone clones and the varieties they determine |
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Authors: | B. A. Davey R. W. Quackenbush D. Schweigert |
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Affiliation: | (1) Department of Mathematics, La Trobe University, 3083 Bundoora, Victoria, Australia;(2) Department of Mathematics and Astronomy, University of Manitoba, R3T2N2 Winnipeg, Manitoba, Canada;(3) Fachbereich Mathematik, Universität Kaiserslautern, D-6750 Kaiserslautern, Germany |
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Abstract: | A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC. |
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Keywords: | 06A10 08A40 08B10 08C05 08C15 |
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