A join-semilattice L with top is said to be conjunctive if every principal ideal is an intersection of maximal ideals. (This is equivalent to a first-order condition in the language of semilattices.) In this paper, we explore the consequences of the conjunctivity hypothesis for L, and we define and study a related property, called “ideal conjunctivity,” which is applicable to join-semilattices without top. Results include the following: (a) Every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact \(T_1\)-topology on \(\mathop {\mathrm {max}}L\), the set of maximal ideals of L, and under weak hypotheses this representation is functorial. (b) Every Wallman base for a topological space is conjunctive; we give an example of a conjunctive annular base that is not Wallman. (c) The free distributive lattice over a conjunctive join-semilattice L is a subsemilattice of the power set of \(\mathop {\mathrm {max}}L\). (d) For an arbitrary join-semilattice L: if every u-maximal ideal is prime (i.e., the complement is a filter) for every \(u\in L\), then L satisfies Katriňák’s distributivity axiom. (This appears to be new, though the converse is well known.) If L is conjunctive, all the 1-maximal ideals of L are prime if and only if L satisfies a weak distributivity axiom due to Varlet. We include a number of applications.

如果每个主理想都是极大理想的交集，则称具有顶部的连接半格L是合取的。（这相当于半格语言中的一阶条件。）在本文中，我们探讨了L的合取性假设的后果，我们定义并研究了一个相关的性质，称为“理想合取性”，它适用于加入没有顶部的半格。结果包括以下内容：（a）对于\(\mathop {\mathrm {max}}L\)上的紧凑\(T_1\) -拓扑，每个联合连接半格同构于连接闭合子基，集合L 的极大理想，并且在弱假设下，这种表示是函子的。(b) 拓扑空间的每个 Wallman 基都是合取的；我们给出了一个不是 Wallman 的连接环形底座的例子。(c) 合取连接半格L 上的自由分配格是\(\mathop {\mathrm {max}}L\)的幂集的亚半格。(d) 对于任意连接半格L：如果每个u -最大理想对于每个\(u\in L\)都是素数（即补码是过滤器），则L满足 Katriňák 的分配公理。（这似乎是新的，但相反的是公知的。）如果大号是结膜，所有的1极大理想大号是素数当且仅当L满足由于 Varlet 引起的弱分配公理。我们包括许多应用程序。