Geodesic
Cover Set Lattices
Canadian journal of mathematics, Tome 32 (1980) no. 5, pp. 1177-1205

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Adams, M. E.; Sichler, J. Cover Set Lattices. Canadian journal of mathematics, Tome 32 (1980) no. 5, pp. 1177-1205. doi: 10.4153/CJM-1980-089-0
@article{10_4153_CJM_1980_089_0,
     author = {Adams, M. E. and Sichler, J.},
     title = {Cover {Set} {Lattices}},
     journal = {Canadian journal of mathematics},
     pages = {1177--1205},
     year = {1980},
     volume = {32},
     number = {5},
     doi = {10.4153/CJM-1980-089-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-089-0/}
}
TY  - JOUR
AU  - Adams, M. E.
AU  - Sichler, J.
TI  - Cover Set Lattices
JO  - Canadian journal of mathematics
PY  - 1980
SP  - 1177
EP  - 1205
VL  - 32
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-089-0/
DO  - 10.4153/CJM-1980-089-0
ID  - 10_4153_CJM_1980_089_0
ER  - 
%0 Journal Article
%A Adams, M. E.
%A Sichler, J.
%T Cover Set Lattices
%J Canadian journal of mathematics
%D 1980
%P 1177-1205
%V 32
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-089-0/
%R 10.4153/CJM-1980-089-0
%F 10_4153_CJM_1980_089_0

[1] 1. Adams, M. E. and Sichler, J., Bounded endomorphisms of lattices of finite height, Can. J. Math. 29 (1977), 1254–1263. Google Scholar

[2] 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] 3. Adams, M. E. and Sichler, J., Subfunctors of full embeddings of algebraic categories, to appear in Alg. Universalis. Google Scholar

[4] 4. Chen, C. C. and Grätzer, G., On the construction of complemented lattices, J. Alg. 11 (1969), 56–63. Google Scholar

[5] 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] 6. Grätzer, G., General lattice theory (Birkhäuser Verlag, 1978). Google Scholar | DOI

[7] 7. Hedrlín, Z. and Pultr, A., On full embeddings of categories of algebras, 111. J. Math. 10 (1966), 392–405. Google Scholar

[8] 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] 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] 10. Hell, P., Full embeddings into some categories of graphs, Alg. Universalis 2 (1972), 129–141. Google Scholar

[11] 11. Jónsson, B., Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121. Google Scholar

[12] 12. Kučera, L. and Pultr, A., Non-algebraic concrete categories, J. Pure Appl. Alg. 3 (1973), 95–102. Google Scholar

[13] 13. Pultr, A. and Trnková, V., Combinatorial, algebraic, and topological representations of categories (North Holland, 1980). Google Scholar

[14] 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

Cité par Sources :