Vol. 35, No. 3, 1970

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
On the endomorphism semigroup (and category) of bounded lattices

George Grätzer and J. Sichler

Vol. 35 (1970), No. 3, 639–647
Abstract

Recently, a large number of papers have been published on the representations of a semigroup with identity (i.e., a momoid) as the endomorphism semigroup of algebras and relational systems of various kinds. For instance Z. Hedrlín and J. Lambek showed that every monoid can be represented as the endomorphism semigroup of a semigroup.

Conspicuously missing from the list of algebras for which theorems of this type can be proved are lattices. The reason for this is that any constant map is a lattice endomorphism, and therefore endomorphism semigroups of lattices are very special. For partially ordered sets one can eliminate constant maps as endomorphisms by considering only those maps φ that satisfy x < y implies xφ < yφ (Z. Hedrlin and R. H. McDowell); however, this would not be a very natural condition to impose on lattice endomorphisms.

The approach of this paperl is to consider bounded lattices only, that is, lattices with smallest element 0 and largest element 1, and as endomorphisms to admit only those lattice endomorphisms that preserve 0 and 1 (i.e., keep 0 and 1 fixed; this amounts to considering 0 and 1 as nullary operations). Such endomorphisms are usually called {0,1}-endomorphisms but they are called simply endomorphisms in this paper.

The first result is that every monoid is isomorphic to the monoid of all endomorphisms of a bounded lattice.

One can also consider lattices with complementation L;,,′⟩, where is a complementation, that is, for every a L,aa= 0 and aa= 1. For such algebras an endomorphism is a lattice endomorphism φ that preserves , that is, ()= afφ for all a L. Every lattice with complementation is bounded, and any such endomorphism preserves 0 and 1.

The second result is that every monoid is isomorphic to the endomorphism semigroup of a lattice with complementation.

Both these results are consequences of much stronger theorems proved in this paper.

Mathematical Subject Classification
Primary: 06.30
Secondary: 20.00
Milestones
Received: 21 April 1970
Published: 1 December 1970
Authors
George Grätzer
J. Sichler