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Bounded Endomorphisms of Lattices of Finite Height

Published online by Cambridge University Press:  20 November 2018

M. E. Adams  and
J. Sichler
M. E. Adams
Affiliation:
University of Manitoba, Winnipeg, Manitoba
J. Sichler
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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Every monoid M is isomorphic to the monoid End0,1(L) of all (0,1)-preserving endomorphisms of a bounded lattice L, see [3]. The lattices L with M ≌ End0,1(L) that are constructed there are of an arbitrary infinite cardinality not smaller than that of M, and they all have infinite chains. The aim of the present article is to supplement these results. It will be shown that every finite monoid M is representable as End0,1(L) of a finite lattice. In addition, an account of the difficulties involved in attempting to characterize endomorphism monoids of lattices of a fixed finite height will be given.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 6 , 01 December 1977 , pp. 1254 - 1263
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Adams, M. E. and Sichler, J., On cover set lattices and their applications, to appear.Google Scholar
2. Grätzer, G., General lattice theory (Birkhâuser Verlag, Basel, 1976).Google Scholar
3. Grätzer, G. and Sichler, J., On the endomorphism semigroup (and category) of bounded lattices, Pacific J. Math. 35 (1970), 639–647.Google Scholar
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5. Sichler, J., Testing categories and strong universality, Can. J. Math. 25 (1973), 370–385.Google Scholar