On the endomorphism monoids of (uniquely) complemented lattices

Authors:
G. Grätzer and J. Sichler

Journal:
Trans. Amer. Math. Soc. **352** (2000), 2429-2444

MSC (1991):
Primary 06B25; Secondary 08B20

Published electronically:
February 14, 2000

MathSciNet review:
1751222

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Let be a lattice with and . An endomorphism of is a *-endomorphism*, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a *uniquely complemented lattice*; moreover, if is finite, then can be chosen as a *finite complemented lattice*.

**1.**M. E. Adams and J. Sichler,*Bounded endomorphisms of lattices of finite height*, Canad. J. Math.**29**(1977), no. 6, 1254–1263. MR**0447059****2.**M. E. Adams, V. Koubek, and J. Sichler,*Homomorphisms and endomorphisms of distributive lattices*, Houston J. Math.**11**(1985), no. 2, 129–145. MR**792189****3.**C. C. Chen and G. Grätzer,*On the construction of complemented lattices*, J. Algebra**11**(1969), 56–63. MR**0232715****4.**R. P. Dilworth,*Lattices with unique complements*, Trans. Amer. Math. Soc.**57**(1945), 123–154. MR**0012263**, 10.1090/S0002-9947-1945-0012263-6**5.**Robert P. Dilworth,*The Dilworth theorems*, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1990. Selected papers of Robert P. Dilworth; Edited by Kenneth P. Bogart, Ralph Freese and Joseph P. S. Kung. MR**1111485****6.**G. Grätzer,*A reduced free product of lattices*, Fund. Math.**73**(1971), 21-27. MR**46:71010****7.**G. Grätzer,*Free products and reduced free products of lattices*, Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973) Dept. Math., Univ. Houston, Houston, Tex., 1973, pp. 539–563. MR**0396351****8.**-,*General Lattice Theory*, Pure and Applied Mathematics**75**, Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New York-London; Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52. Birkhäuser Verlag, Basel-Stuttgart; Akademie Verlag, Berlin, 1978. xiii+381 pp. MR**80c:06001a,b****9.**-,*General Lattice Theory. Second Edition,*Birkhäuser Verlag, Basel, 1998. xix+663 pp. CMP**99:07****10.**G. Grätzer and J. Sichler,*On the endomorphism semigroup (and category) of bounded lattices*, Pacific J. Math.**35**(1970), 639–647. MR**0277442****11.**-,*On the endomorphism monoid of complemented lattices*, AMS Abstract 97T-06-98.**12.**V. Koubek and J. Sichler,*Universality of small lattice varieties*, Proc. Amer. Math. Soc.**91**(1984), no. 1, 19–24. MR**735556**, 10.1090/S0002-9939-1984-0735556-3**13.**H. Lakser,*Simple sublattices of free products of lattices*, Abstract, Notices Amer. Math. Soc.**19**(1972), A 509.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
06B25,
08B20

Retrieve articles in all journals with MSC (1991): 06B25, 08B20

Additional Information

**G. Grätzer**

Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg MB R3T 2N2, Canada

Email:
gratzer@cc.umanitoba.ca

**J. Sichler**

Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg MB R3T 2N2, Canada

Email:
sichler@cc.umanitoba.ca

DOI:
https://doi.org/10.1090/S0002-9947-00-02628-3

Keywords:
Endomorphism monoid,
complemented lattice,
uniquely complemented lattice

Received by editor(s):
May 28, 1997

Published electronically:
February 14, 2000

Additional Notes:
The research of both authors was supported by the NSERC of Canada.

Article copyright:
© Copyright 2000
American Mathematical Society