The Factorization and Representation of Lattices
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 ⩽̸ Lb. For a jm-lattice 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.
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
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)
Article - Journal
© 1975 American Mathematical Society, All rights reserved.