The factorization and representation of lattices
Abstract: For a complete lattice , in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say is a jm-lattice) we define the poset of irreducibles to be the poset (of height one) is the set of completely join-irreducibles and is the set of completely meet-irreducibles) ordered as follows: if and only if , and . For a jm-lattice , the automorphism groups of and are isomorphic, can be reconstructed from , and the irreducible factorization of can be gotten from the components of . 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 ). Thus 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 for some jm-lattice . We also characterize those posets of height 1 which are for a completely distributive jm-lattice, as well as those posets which are for some geometric lattice .
More generally, if is a complete lattice, many of the above arguments apply if we use ``join-spanning'' and ``meet-spanning'' subsets of , instead of and . If is an arbitrary lattice, the same arguments apply to ``join-generating'' and ``meet-generating'' subsets of .
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Keywords: Poset of irreducibles, completely join-irreducible, Galois connection, irreducible factorization, representations, group of automorphism, geometric lattice, poset of join-irreducibles, join-spanning set, distributive lattice, separators, center
Article copyright: © Copyright 1975 American Mathematical Society