Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. M.E. Adams and J. Sichler, Bounded endomorphisms of lattices of finite height, Canad. J. Math. 24 (1977), 1254-1263. MR 56:5374
  • 2. M.E. Adams, V. Koubek, and J. Sichler, Homomorphisms and endomorphisms of distributive lattices, Houston J. Math. 11 (1985), 129-145. MR 86m:06016
  • 3. C.C. Chen and G. Grätzer, On the construction of complemented lattices, J. Algebra 11 (1969), 56-63. MR 38:1038
  • 4. R.P. Dilworth, Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945), 123-154. MR 7:1b
  • 5. -, The Dilworth theorems. Selected papers of Robert P. Dilworth, Edited by Kenneth P. Bogart, Ralph Freese, and Joseph P.S. Kung. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1990. xxvi+465 pp. MR 92f:01002
  • 6. G. Grätzer, A reduced free product of lattices, Fund. Math. 73 (1971), 21-27. MR 46:71010
  • 7. -, Free products and reduced free products of lattices, Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973), pp. 539-563. Dept. Math., Univ. Houston, Houston, Tex. MR 53:218
  • 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 43:3175
  • 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), 19-24. MR 85m:06015
  • 13. H. Lakser, Simple sublattices of free products of lattices, Abstract, Notices Amer. Math. Soc. 19 (1972), A 509.

Similar Articles

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

American Mathematical Society