Document Type
Article
Publication Date
8-24-2021
Abstract
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 T1-topology on 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 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 ϵ 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.
Recommended Citation
C. Delzell, O. Ighedo & J. Madden: Conjunctive Join-Semilattices. Algebra Univer., (2021), 82:51. https://doi.org/10.1007/s00012-021-00744-3
Peer Reviewed
1
Copyright
Springer
Comments
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Algebra Universalis, volume 51, in 2021 following peer review. The final publication may differ and is available at Springer via https://doi.org/10.1007/s00012-021-00744-3.
A free-to-read copy of the final published article is available here.