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On the Structure of Finitely Presented Lattices

Published online by Cambridge University Press:  20 November 2018

G. Gratzer ,
A. P. Huhn  and
H. Lakser
G. Gratzer
Affiliation:
University of Manitoba, Winnipeg, Manitoba
A. P. Huhn
Affiliation:
University of Manitoba, Winnipeg, Manitoba
H. Lakser
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)

It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.

In this paper we prove a structure theorem that could be said to verify this conjecture.

THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 404 - 411
Copyright
Copyright © Canadian Mathematical Society 1981

References

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