The factorization and representation of lattices
The factorization and representation of lattices
复制标题
格的因式分解和表示
DOI：
10.1090/s0002-9947-1975-0360386-3
复制
发表时间：
1975
影响因子：
1.3
通讯作者：
G. Markowsky
中科院分区：
文献类型：
--
作者：
G. Markowsky
对于完全格L，其中每个元素都是完全并不可约的并和完全相遇不可约的交（我们说L是一个fm-格）我们定义不可约偏序集P（L）为偏序集（高度为1）J（L）U M（L）（J（L）是完全并不可约的集合，M（L）是完全交不可约的集合），顺序如下：a < P（L）B当且仅当a ∈ J（L），B ∈ M（L），a %L B.对于一个jm-格L，L的自同构群与P（L）同构，L可以由P（L）重构，L的不可约因子分解可以由P（L）的分支得到.事实上，我们可以根据它的分离子（或P（L）的连通分支的并集）给出jm-格的中心的简单特征。因此P（L）将有限分配格的并不可约偏序集的许多性质推广到所有的jm-格类。我们刻画了某些jm格L的高为1的偏序集P（L）。我们还刻画了对于完全分配的jm-格，高度为1的偏序集为P（L），以及对于某些几何格L，高度为1的偏序集为P（L）。更一般地说，如果L是一个完备格，如果我们使用L的“join-spanning”和“meet-spanning”子集，而不是J（L）和M（L），则上述许多论点都适用。如果L是一个任意格，同样的论证适用于L的“并生成”和“交生成”子集。本文讨论了用高为1的偏序集上的闭包算子表示格的问题。两个集合R S X x Y之间的每一个关系都在X的幂集和Y的幂集之间导出一个伽罗瓦联络，因此确定了一个闭集的格L（R）（比如说在X中）。设L是有限格，X和Y是L的并可约元和半可约元的集合，R是关系t，则L L（R）.这个想法平凡地延伸到完整的infmite格，其中每个元素都是一个接收的编辑1973年8月17日，并在修订的形式，1974年1月15日。AMS（MOS）主题分类（1970年）。初级06 A15、06 A20、06 A23、06 A35;次级06 A30、06 A45。
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 fm-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-irreducibles and M(L) is the set of completely meet-irreducibles) ordered as follows: a < P(L) b if and only if a E J(L), b E M(L), and a % 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. 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 S 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 joinand meetirreducible elements of L, and R is the relation t, then L L(R). This idea extends trivially to complete infmite lattices in which every element is a Received by the editors August 17, 1973 and, in revised form, January IS, 1974. AMS (MOS) subject classifications (1970). Primary 06A15, 06A20, 06A23, 06A35; Secondary 06A30, 06A45.