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

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

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*.

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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