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Universal Varieties Of (0, 1)-Lattices

Published online by Cambridge University Press:  20 November 2018

P. Goralčík ,
V. Koubek  and
J. Sichler
P. Goralčík
Affiliation:
MFF KU Sokolovská 83 186 00 Praha 8 Czechoslovakia
V. Koubek
Affiliation:
MFF KU Malostranská nám. 25 118 00 Praha 1 Czechoslovakia
J. Sichler
Affiliation:
Department of Mathematics University of Manitoba Winnipeg, Manitoba Canada R3T 2N2
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This article fully characterizes categorically universal varieties of (0, 1)-lattices (that is, lattices with a least element 0 and a greatest element 1 regarded as nullary operations), thereby concluding a series of partial results [3, 5, 8, 10, also 14] which originated with the proof of categorical universality for the variety of all (0, 1)-lattices by Grätzer and Sichler [6].

A category C of algebras of a given type is universal if every category of algebras (and equivalently, according to Hedrlín and Pultr [7 or 14], also the category of all graphs) is isomorphic to a full subcategory of C. The universality of C is thus equivalent to the existence of a full embedding Φ : G→C of the category G of all graphs and their compatible mappings into C. When Φ assigns a finite algebra to every finite graph, we say that C is finite-to-finite universal.

Keywords

06B2008A3518B15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 3 , 01 June 1990 , pp. 470 - 490
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Adams, M.E., Koubek, V. and Sichler, J., Homomorphisms and endomorphisms of distributivelattices, Houston J. Math. 11 (1985), 129–145.Google Scholar
2. Adams, M.E. and Sichler, J., Bounded endomorphisms of lattices of finite height, Canad. J. Math. 29 (1977), 163–182.Google Scholar
3. Adams, M.E. and Sichler, J., Cover set lattices, Canad. J. Math. 32 (1980), 1177–1205.Google Scholar
4. Birkhoff, G., On groups of automorphisms, Rev. Un. Mat. Argentina 11 (1946), 155–157.Google Scholar
5. Goralčfk, P., Koubek, V. and Prohle, P., A universality condition for varieties of 0,1 -lattices, Colloq. Math. Soc. J. Bolyai, 43. Lectures in Universal Algebra, Szeged (Hungary), 1983, North-Holland, Amsterdam, 1985, 143–154.Google Scholar
6. Grätze, G. and Sichler, J., On the endomorphism semigroup (and category) of bounded lattices,, Pacific J. Math. 35 (1970), 639–647.Google Scholar
7. Hedrlín, Z. and Pultr, A., On full embeddings of categories of algebras, Illinois J. Math. 10 (1966), 392–406.Google Scholar
8. Koubek, V., Towards minimal binding varieties of lattices, Canad. J. Math. 36 (1984), 263–285.Google Scholar
9. Koubek, V., Infinite image homomorphisms of distributive bounded lattices, Colloq. Math. Soc. János Bolyai 43, Lectures in Universal Algebra, Szeged (Hungary) 1983, North-Holland, Amsterdam, 1985, 241–281.Google Scholar
10. Koubek, V. and Sichler, J., Universality of small lattice varieties, Proc. Amer. Math. Soc. 91 (1984), 19–24.Google Scholar
11. McKenzie, R. and Tsinakis, C., On recovering bounded distributive lattice from its endomorphismmonoid, Houston J. Math. 7 (1981), 525–529.Google Scholar
12. Priestley, H.A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186–190.Google Scholar
13. Priestley, H.A., Ordered sets and duality for distributive lattices, Ann. Discrete Math. 23 (1984), 39–60.Google Scholar
14. Pultr, A. and Trnková, V., Combinatorial, algebraic and topological representations of groups, semigroups and categories, North-Holland, Amsterdam, 1980.Google Scholar
15. Sichler, J., Non-constant endomorphisms of lattices, Proc. Amer. Math. Soc. 34 (1972), 67–70.Google Scholar