Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T00:59:57.475Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

References

1. Adams, M. E. and Sichler, J., Bounded endomorphisms of lattices of finite height, Can. J. Math. 29 (1977), 1254–1263.Google Scholar
2. Adams, M. E. and Sichler, J., Hotnomorphisnts of bounded lattices with a given sublattice, Arch. Math. (Basel) 30 (1978), 122–128.Google Scholar
3. Adams, M. E. and Sichler, J., Subfunctors of full embeddings of algebraic categories, to appear in Alg. Universalis.Google Scholar
4. Chen, C. C. and Grätzer, G., On the construction of complemented lattices, J. Alg. 11 (1969), 56–63.Google Scholar
5. Grätzer, G. and Sichler, J., On the endomorphism semigroup (and category) of bounded lattices, Pacific J. Math. 35 (1970), 639–647.Google Scholar
6. Grätzer, G., General lattice theory (Birkhäuser Verlag, 1978).CrossRefGoogle Scholar
7. Hedrlín, Z. and Pultr, A., On full embeddings of categories of algebras, 111. J. Math. 10 (1966), 392–405.Google Scholar
8. Hedrlín, Z. and Sichler, J., Any boundable binding category contains a proper class of mutually disjoint copies of itself, Alg. Universalis 1 (1971), 97–103.Google Scholar
9. Hedrlín, Z., Extensions of structures and full embeddings of categories, in Proc. Intern. Congr. of Mathematicians, Nice (Gauthier-Villars, Paris, 1971).Google Scholar
10. Hell, P., Full embeddings into some categories of graphs, Alg. Universalis 2 (1972), 129–141.Google Scholar
11. Jónsson, B., Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121.Google Scholar
12. Kučera, L. and Pultr, A., Non-algebraic concrete categories, J. Pure Appl. Alg. 3 (1973), 95–102.Google Scholar
13. Pultr, A. and Trnková, V., Combinatorial, algebraic, and topological representations of categories (North Holland, 1980).Google Scholar
14. Vopěnka, P., Pultr, A., and Hendrlin, Z., A rigid relation exists on any set, Comment. Math. Univ. Carolinae 6 (1965), 149–155.Google Scholar