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Doubling Constructions in Lattice Theory

Published online by Cambridge University Press:  20 November 2018

Alan Day
Alan Day*
Affiliation:
Department of Mathematical Sciences Lakehead University Thunder Bay, OntarioCanada P7B 5E1
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Abstract

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This paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.

Keywords

06B2506B05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 2 , 01 April 1992 , pp. 252 - 269
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Day, A., A simple solution of the word problem for lattices, Can. Math. Bull. 13(1970), 253–254.Google Scholar
2. Day, A., Splitting lattices generate all lattices, Alg. Univ. 7(1977), 163–170.Google Scholar
3. Day, A., Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices, Can. J. Math. 31(1979), 69–78.Google Scholar
4. Day, A., Distributive lattices with finite projective covers, Pacific J. Math 81(1979), 45–59.Google Scholar
5. Day, A. and Ch. Herrmann, Gluings of modular lattices, Order 5(1988), 85–101.Google Scholar
6. Dean, R., Free lattices generated by partially ordered sets and preserving bounds, Can. J. Math. 16(1964), 136–148.Google Scholar
7. Freese, R. Finitely presented lattices: canonical forms and the covering relation Trans. Amer. Math. Soc. 312(1989), 841–860.Google Scholar
8. Freese, R. and Nation, J.B., Covers in free lattices, Trans. Amer. Math. Soc. 288(1985), 1–42.Google Scholar
9. Jezek, J. and Slavík, V., Free lattices over join-trivial partial lattices, Alg. Univ. 27(1990), 10–31.Google Scholar
10. Kostinsky, A., Projective lattices and bounded homomorphisms, Pacific J. Math. 40(1972), 111–119.Google Scholar
11. Kisielewicz, A., A solution of Dedkind's problem on the number of isotone Boolean functions, J. Reine Angew. Math., 386(1988), 139–144.Google Scholar
12. McKenzie, R., Equational bases and non-modular lattice varieties, Trans. Amer. Math. Soc. 174(1972), 1–43.Google Scholar
13. Quackenbush, R., Dedekind'sproblem, Order 2(1986), 415–417.Google Scholar