# The Factorization and Representation of Lattices

## Abstract

For a complete lattice *L*, in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say *L* is a *jm-lattice*) we define the *poset of irreducibles P(L)* to be the poset (of height one) *J(L)* ⊍ *M(L) (J(L)* is the set of completely join-irreducibles and *M(L)* is the set of completely meet-irreducibles) ordered as follows: α < * _{P(L)}b* if and only if α ∈

*J(L), b*∈

*M(L)*, and α ⩽̸

*. For a jm-lattice*

_{L}b*L*, the automorphism groups of

*L*and

*P(L)*are isomorphic,

*L*can be reconstructed from

*P(L)*, and the irreducible factorization of

*L*can be gotten from the components of

*P(L)*. In fact, we can give a simple characterization of the center of a jm-lattice in terms of its separators (or unions of connected components of

*P(L)*). Thus

*P(L)*extends many of the properties of the poset of join-irreducibles of a finite distributive lattice to the class of all jm-lattices.

We characterize those posets of height 1 which are

*P(L)*for some jmlattice

*L*. We also characterize those posets of height 1 which are

*P(L)*for a completely distributive jm-lattice, as well as those posets which are

*P(L)*for some geometric lattice

*L*.

More generally, if

*L*is a complete lattice, many of the above arguments apply if we use "join-spanning" and "meet-spanning" subsets of

*L*, instead of

*J(L)*and

*M(L)*. If

*L*is an arbitrary lattice, the same arguments apply to "join-generating" and "meet-generating" subsets of

*L*.

## Recommended Citation

G.
Markowsky,
"The Factorization and Representation of Lattices," *Transactions of the American Mathematical Society*, vol. 203, no. 476, pp. 185 - 200, American Mathematical Society, Mar 1975.

The definitive version is available at https://doi.org/10.1090/S0002-9947-1975-0360386-3

## Department(s)

Computer Science

## Keywords and Phrases

Center; Completely join-irreducible; Distributive lattice; Galois connection; Geometric lattice; Group of automorphism; Irreducible factorization; Join-spanning set; Poset of irreducibles; Poset of join-irreducibles; Representations; Separators

## International Standard Serial Number (ISSN)

0002-9947

## Document Type

Article - Journal

## Document Version

Citation

## File Type

text

## Language(s)

English

## Rights

© 1975 American Mathematical Society, All rights reserved.

## Publication Date

01 Mar 1975