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) $J(L) \cup M(L)(J(L)$ is the set of completely join-irreducibles and $M(L)$ is the set of completely meet-irreducibles) ordered as follows: $a{ < _{P(L)}}b$ if and only if $a \in J(L),b \in M(L)$, and $a \nleqslant { _L}b$. 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 jm-lattice $L$. 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 $L$.