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 \(T_1\)-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.
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Acknowledgements
We thank the anonymous referee for thorough and influential reports on the initial and subsequent versions of this paper and for generously inviting us to include his/her contributions, particularly in subsection 4.2. By directing our attention to Varlet’s works [24, 25], the referee helped to bring about significant extensions of the results we originally claimed. Other remarks by the referee led to many additional improvements throughout the paper. Our thanks go to the Department of Mathematics at Louisiana State University for sustaining a supportive environment for research during the COVID pandemic.
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Communicated by Presented by W. Wm. McGovern.
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Ighedo was supported during her 2019-2020 sabbatical at LSU by research grants from the Oppenheimer Memorial Trust in South Africa (Grant Number 21461/01), the University Capacity Development Programme in South Africa (Grant Number 365), and the College of Science, Engineering and Technology of the University of South Africa.
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Delzell, C.N., Ighedo, O. & Madden, J.J. Conjunctive join-semilattices. Algebra Univers. 82, 51 (2021). https://doi.org/10.1007/s00012-021-00744-3
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DOI: https://doi.org/10.1007/s00012-021-00744-3
Keywords
- Join-semilattice
- Conjunctivity
- Distributive join-semilattice
- Maximal ideal
- Prime ideal
- Topological representation
- Wallman base