合取并半格
Conjunctive Join Semi-Lattices
摘要：
一个并半格$L$被称为合取的，如果它有一个顶元$1$，并且它满足以下一阶条件：对于任何两个不同的$a,b\in L$，存在$c\in L$使得$a\vee c\not=1=b\vee c$或$a\vee c=1\not=b\vee c$。等价地，一个并半格是并的，如果每个主理想都是极大理想的交。我们给出了一些简单的例子，证明了一个合取并半格可能没有任何素理想。（联半格的极大理想不一定是素的。证明了在$L$的极大理想集$\mathrm{Max} L$上，紧$T_1$ -拓扑的每一个合取并半格同构于一个并闭子基.该表示是规范的，当应用到一个连接封闭的子基的紧凑$T_1$ -空间$X$，空间产生的表示是同胚的$X$。我们称一个并半格态射$ϕ:L\to M$是合取的，如果$ϕ^{-1}(w)$是$L$的极大理想的交，只要$w$是$M$的极大理想。我们证明了从$\mathrm{Max} M$到$\mathrm{Max} L$的多值函数诱导了合取并半格之间的每一个合取态射。一个拓扑空间的基如果是一个格就称为环基，如果它是环基，并且对于任何基本开$U$中的任何点$u$，都有一个基本开$V$，它不包含$u$，并且与$U$一起覆盖$X$。很容易证明每一个沃尔曼基都是合取的。我们给出了一个例子，合取环形基地，不是沃尔曼。最后，我们研究了合取连接半格$L$上的自由分配格$dL$。在一般情况下，它是不合取的，但我们表明，一定的规范，代数定义的商的$dL$是同构的子格的拓扑结构的表示空间是由$L$。我们描述了许多应用程序。
英文摘要：
A join-semilattice $L$ is said to be conjunctive if it has a top element $1$ and it satisfies the following first-order condition: for any two distinct $a,b\in L$, there is $c\in L$ such that either $a\vee c\not=1=b\vee c$ or $a\vee c=1\not=b\vee c$. Equivalently, a join-semilattice is conjunctive if every principal ideal is an intersection of maximal ideals. We present simple examples showing that a conjunctive join-semilattice may fail to have any prime ideals. (Maximal ideals of a join-semilattice need not be prime.) We show that every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact $T_1$-topology on $\mathrm{Max} L$, the set of maximal ideals of $L$. The representation is canonical in that when applied to a join-closed subbase for a compact $T_1$-space $X$, the space produced by the representation is homeomorphic with $X$. We say a join-semilattice morphism $ϕ:L\to M$ is conjunctive if $ϕ^{-1}(w)$ is an intersection of maximal ideals of $L$ whenever $w$ is a maximal ideal of $M$. We show that every conjunctive morphism between conjunctive join-semilattices is induced by a multi-valued function from $\mathrm{Max} M$ to $\mathrm{Max} L$. A base for a topological space is said to be annular if it is a lattice, and Wallman if it is annular and for any point $u$ in any basic open $U$, there a basic open $V$ that misses $u$ and together with $U$ covers $X$. It is easy to show that every Wallman base is conjunctive. We give an example of a conjunctive annular base that is not Wallman. Finally, we examine the free distributive lattice $dL$ over a conjunctive join semilattice $L$. In general, it is not conjunctive, but we show that a certain canonical, algebraically-defined quotient of $dL$ is isomorphic to the sub-lattice of the topology of the representation space that is generated by $L$. We describe numerous applications.
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