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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 Katriňak’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.

Convex geometry is a closure space $$(G,\phi )$$ with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary and sufficient conditions on a closure operator $$\phi $$ of convex geometry $$(G,\phi )$$ so that its convex dimension equals 2, equivalently, they are represented by segments on a line. These conditions, for a given convex geometry $$(G,\phi )$$, can be checked in polynomial time in two parameters: the size of the base set |G| and the size of the implicational basis of $$(G,\phi )$$.

In this paper, we show that for any finite idempotent non-congruence 3-permutable algebra \(\mathbf {A}\), it is always the case that \({{\,\mathrm{HSP}\,}}(\mathbf {A})\) contains an algebra carrying a binary reflexive compatible relation having a certain shape, which we have called a 2-dimensional special Hagemann relation with middle part. As a result of this property, we are able to show that the join of any pair of locally finite idempotent non-congruence 3-permutable varieties in the lattice of interpretability types, fails to be congruence 3-permutable, yielding a primeness argument for this specific setting.

The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators ◊ and □ on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending joins to meets in the first argument and meets to meets in the second). This paper introduces a refinement of the topological duality of Part I that makes explicit the topological distinction between the duals of meet homomorphisms and of join homomorphisms. As a result, quasioperators can be represented by certain continuous maps on the topological duals.

The “Modularity Conjecture” is the assertion that the join of two nonmodular varieties in the lattice of interpretability types is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning n-permutability for some n, and the satisfaction of nontrivial congruence identities.

We generalise in three different directions two well-known results in universal algebra. Gratzer, Lakser and Plonka proved that independent subvarieties $${\mathcal{V}_{1}, \mathcal{V}_{2}}$$ of a variety $${\mathcal{V}}$$ are disjoint and such that their join $${\mathcal{V}_{1} \vee \mathcal{V}_{2}}$$ (in the lattice of subvarieties of $${\mathcal{V}}$$ ) is their direct product $${\mathcal{V}_{1} \times \mathcal{V}_{2}}$$ . Jonsson and Tsinakis provided a partial converse to this result: if $${\mathcal{V}}$$ is congruence permutable and $${\mathcal{V}_{1}, \mathcal{V}_{2}}$$ are disjoint, then they are independent (and so $${\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}$$ ). We show that (i) if $${\mathcal{V}}$$ is subtractive, then Jonsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if $${\mathcal{V}}$$ satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and $${\mathcal{V}_{1} \vee \mathcal{V}_{2}}$$ is the subdirect product of $${\mathcal{V}_{1}}$$ and $${\mathcal{V}_2}$$ ; (iii) the same holds if $${\mathcal{V}}$$ is congruence 3-permutable.

In this paper we show that, given a complete lattice $$ \mathcal{L} $$ , the following three lattices are the same: (1) the lattice of closure relations on $$ \mathcal{L} $$ , (2) the lattice of meet-closed subsets of $$ \mathcal{L} $$ , and (3) the lattice of complete join congruence relations on $$ \mathcal{L} $$ .

We define the $ \scr A $ -ideals of a poset – or equally of a quasi-ordered set – for various collections of subsets $ \scr A $ and corresponding $ \scr A $ -ideal continuity for functions. This leads us to a choice-free $ \scr A $ -ideal continuous imbedding of a poset into a $ \scr B $ -join complete poset with an appropriate universal mapping property. Topological applications include the imbedding of Scott spaces and Alexandrov spaces into up-complete Scott spaces.

For a complete lattice C, we consider the problem of establishing when the complete lattice of complete congruence relations on C is a complete sublattice of the complete lattices of join- or meet-complete congruence relations on C. We first argue that this problem is not trivial, and then we show that it admits an affirmative answer whenever C is continuous for the join case and, dually, co-continuous for the meet case. As a consequence, we prove that if C is continuous then each principal filter generated by a continuous complete congruence on C is pseudocomplemented.

We associate to every hypergraphГ a commutative quasi-hypergroupH qG and find a necessary and sufficient condition onГ so thatH Г is associative. For certainГ, any finiteГ included, we determine a sequenceГ=Г 0, Г1,⋯, Гn of hypergraphs such that ifH 0 ,H 1 ,H⋯, H n is the sequence of the associated quasi-hypergroups,H n is a join space.

In this paper we study local Noether lattices in which the maximal element is join-principal or weak join-principal. A characterization of maximal join-principal elements involving distributivity and independence is given and the distributive local Noether lattices with weak join-principal maximal elements are determined.

Although equationally compact semilattices have been completely characterized [4], the question of J. Mycielski "Is every equationally compact semilattice the retract of a compact topological semilattice?" has heretofore remained unanswered. The main purpose of the present paper is to provide an affirmative answer to this question. Further, a new notion of "algebraic" compactness is introduced which among all semilattices singles out exactly those in which every chain is finite. Such semilattices are in turn compact topological ones in view of the more general result that the class of compact topological semilattices includes all join-complete semilattices in which every chain has a least element. Throughout this paper the term "semilattice" shall mean "join semilattice". The results presented here form a part of the author's doctoral thesis. For inspiration and guidance during the course of this investigation the author expresses gratitude to G.H. Wenzel.