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Finitely generated pseudocomplemented distributive lattices

Published online by Cambridge University Press:  09 April 2009

J. Berman and
ph. Dwinger
J. Berman
Affiliation:
University of Illinois at ChicagoChicago, Illinois, U. S. A.
ph. Dwinger
Affiliation:
University of Illinois at ChicagoChicago, Illinois, U. S. A.
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If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 2 , March 1975, pp. 238 - 246
Copyright
Copyright © Australian Mathematical Society 1975

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