The factorization and representation of lattices

Abstract

For a complete lattice L L , in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say L L is a jm-lattice) we define the poset of irreducibles P ( L ) P(L) to be the poset (of height one) J ( L ) ∪ M ( L ) ( J ( L ) J(L) \cup M(L)(J(L) is the set of completely join-irreducibles and M ( L ) M(L) is the set of completely meet-irreducibles) ordered as follows: a > P ( L ) b a{ > _{P(L)}}b if and only if a ∈ J ( L ) , b ∈ M ( L ) a \in J(L),b \in M(L) , and a ⪇ L b a \nleqslant { _L}b . For a jm-lattice L L , the automorphism groups of L L and P ( L ) P(L) are isomorphic, L L can be reconstructed from P ( L ) P(L) , and the irreducible factorization of L L can be gotten from the components of P ( L ) 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 ) P(L) ). Thus P ( L ) 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 ) P(L) for some jm-lattice L L . We also characterize those posets of height 1 which are P ( L ) P(L) for a completely distributive jm-lattice, as well as those posets which are P ( L ) P(L) for some geometric lattice L L . More generally, if L L is a complete lattice, many of the above arguments apply if we use “join-spanning” and “meet-spanning” subsets of L L , instead of J ( L ) J(L) and M ( L ) M(L) . If L L is an arbitrary lattice, the same arguments apply to “join-generating” and “meet-generating” subsets of L L .