Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-29T10:21:46.075Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

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

References

Balbes, R. (1969), ‘A representation theory for prime and implicativde semilattices’, Trans. Amer. Math. Soc. 136, 261–267.CrossRefGoogle Scholar
Balbes, R. (1973), ‘On free pseudo-c'omplemented and relatively pseudo-complemented semilattices’, Fund. Math. 8, 119–131.CrossRefGoogle Scholar
Dyson, V. (1965), An embedding theorem for semilattices, (Hughes Aircraft Technical Report, 1965).Google Scholar
Frink, O. (1962), ‘Pseudo-complements in semi-lattices’, Duke Math. J. 29, 505–514.CrossRefGoogle Scholar
Grätzer, G. (1968), Universal Algebra, (Van Nostrand, 1968).Google Scholar
Grätzer, G. (1971), Lattice Theory, (W. H. Freeman and Co., 1971).Google Scholar
Lakser, H. (1971), ‘The structure of pseudo-complemented, distributive lattices I. Subdirect decomposition’, Trans. Amer. Math. Soc. 156, 335–342.Google Scholar
Lakser, H. (1973), ‘Principal congruences of pseudocomplemented distributive lattices’, Proc. Amre. Math. Soc. 37, 32–36.CrossRefGoogle Scholar
Lee, K. B. (1970), ‘Equational classes of distributive pseudocomplemented lattices’, Canad. J. Math. 22 881–891.CrossRefGoogle Scholar
Marczewski, E. (1958), ‘A general scheme of the notions of independence in mathematics’, Bull. Acad. Polon. Sci. 6 731–736.Google Scholar
Urquhart, A. (1973), ‘Free distributive pseudo-complemented lattices’, Algebra Universalis 3, 13–15.CrossRefGoogle Scholar