TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 203, 1975 THE FACTORIZATION AND REPRESENTATION OF LATTICES BY GEORGE MARKOWSKY(') 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) U M(L) (J(L) is the set of completely join-irreduc- ibles and M(L) is the set of completely meet-irreducibles) ordered as follows: a < p/[\ b if and only if a 6 J(L), b e M(L), and a ^C ¡J>. For a jm-lattice L, the automorphism groups of L and P(L) are isomorphic, L can be re- constructed 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. This paper concerns the problem of representing lattices by means of closure operators on partially ordered sets of height 1. Every relation between two sets R <X x Y induces a Galois connection between the power set of X and the power set of Y, and hence determines a lattice L(R) of closed sets (in X, say). If L is a finite lattice, and X and Y are the sets of join-and meet- irreducible elements of L, and R is the relation ^E, then L = L(R). This idea extends trivially to complete infinite lattices in which every element is a Received by the editors August 17, 1973 and, in revised form, January 15, 1974. AMS (MOS) subject classifications (1970). Primary 06A15, 06A20, 06A23, 06A35; Secondary 06A30, 06A45. Key words and phrases. Poset of irreducibles, completely join-irreducible, Galois con- nection, irreducible factorization, representations, group of automorphism, geometric lattice, poset of join-irreducibles, join-spanning set, distributive lattice, separators, center. (') The results described here are partly contained in the author's doctoral thesis [5] which was partly supported by ONR Contract N00014-67-A-0298-0015. Copyright © 1975, American Mathematical Society 185 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use