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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
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Abstract:

Let $L$ be a lattice with $0$ and $1$. An endomorphism $\varphi$ of $L$ is a $\{0,1\}$-endomorphism, if it satisfies $0\varphi = 0$ and $1\varphi = 1$. The $\{0,1\}$-endomorphisms of $L$ form a monoid. In 1970, the authors proved that every monoid $\mathcal M$ can be represented as the $\{0,1\}$-endomorphism monoid of a suitable lattice $L$ with $0$ and $1$. In this paper, we prove the stronger result that the lattice $L$ with a given $\{0,1\}$-endomorphism monoid $\mathcal M$ can be constructed as a uniquely complemented lattice; moreover, if $\mathcal M$ is finite, then $L$ 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