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Cover Set Lattices

Published online by Cambridge University Press:  20 November 2018

M. E. Adams  and
J. Sichler
M. E. Adams
Affiliation:
S.U.N. Y. at New Paltz, New Paltz, New York
J. Sichler
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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The proof of a main result in [1] concerning (0,1)-endomorphisms of finite lattices is based on properties of lattices A(G) derived from the system of independent sets of an undirected loop-free graph G. For a number of questions naturally arising from [1] and [2], however, constructions employing only graph-induced complementation and properties of the lattices A (G) associated with these are no longer adequate. The present paper introduces cover set lattices (a generalization of the lattices A(G)) to deal with some of these questions. A special case of the main result presented here states that for every (0, 1)-lattice L and any monoid homomorphism φ:M → End0,1(L) there exists a lattice K containing L as a (0, 1)-sublattice in such a way that the monoid End0,1(K) of all (0, 1)-endomorphisms of K is isomorphic to M, and the restriction to L of every (0, 1)-endomorphism m of K is the (0, 1)-endomorphism φ(m) of L.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1177 - 1205
Copyright
Copyright © Canadian Mathematical Society 1980

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