Distributive Algebraic Lattice Research Articles

Evaporation schemes in congruence lattices of majority algebras

We investigate the problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a majority algebra. We try the method used in Wehrung’s solution of Dilworth’s Congruence Lattice Problem. For this purpose we introduce the

A note on the algebraicity of L-fuzzy subalgebras in universal algebra

Given a universal algebra $${\mathcal {A}}:=(A;~F^A)$$ of type $${\mathcal {F}}$$, it is well known that the lattice $${\mathbb {S}}ub({\mathcal {A}})$$ of subuniverses of $${\mathcal {A}}$$ is algebraic and its compact elements are exactly finitely generated subuniverses of $${\mathcal {A}}$$. In this paper, under a distributive algebraic lattice $${\mathcal {L}}:=(L;~\wedge ,~\vee ;~0,~1)$$, we characterize the compact elements of the lattice $${\mathbb {F}}s({\mathcal {A}},L)$$ of L-fuzzy subalgebras of $${\mathcal {A}}$$, which is an extension of $${\mathbb {S}}ub({\mathcal {A}})$$ and show that the latter is algebraic.

A completion for distributive nearlattices

The aim of this article is to propose an adequate completion for distributive nearlattices. We give a proof of the existence of such a completion through a representation theorem, which allows us to prove that this completion is a completely distributive algebraic lattice. We show several properties about this completion, and we present a connection with the free distributive lattice extension of a distributive nearlattice. Finally, we consider how can be extended n-ary operations on distributive nearlattices, and we study the basic properties of these extensions.

Completely distributive completions of posets

A $$\Delta_1$$ -completion of a poset is a completion for which, simultaneously, every element is reachable as a join of meets and a meet of joins from the original poset. We focus our attention on $$\Delta_1$$ -completions that can be obtained from polarities $$\langle\mathcal{F},\mathcal{I},\mathcal{R}\rangle$$ where $$\mathcal{F}$$ is a collection of upsets containing the principal upsets, $$\mathcal{I}$$ is a collection of downsets containing the principal downsets of the original poset, and $$R\subseteq\mathcal{F}\times\mathcal{I}$$ is the relation of nonempty intersection. These $$\Delta_1$$ -completions are called $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions, and they satisfy a compactness property. In this paper, we show that if a pair $$\langle\mathcal{F},\mathcal{I}\rangle$$ satisfies a separating condition (similar to the Prime Filter Theorem for distributive lattices), then the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of the original poset is a completely distributive algebraic lattice. Given a poset P and an algebraic closure system $$\mathcal{F}$$ of upsets of P satisfying a distributivity condition, we show how to choose a collection of downsets $$\mathcal{I}$$ of P such that the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of P is a completely distributive algebraic lattice. Then, we study the extensions of additional operations on posets to their corresponding $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions. Finally, we use the previous results to obtain adequate $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions for the classes of Tarski algebras and Hilbert algebras, and for the classes of algebras that are canonically associated (in the sense of abstract algebraic logic) with some propositional logics.

A Spectral-style Duality for Distributive Posets

In this paper, we present a topological duality for a category of partially ordered sets that satisfy a distributivity condition studied by David and Erne. We call these posets mo-distributive. Our duality extends a duality given by David and Erne because our category of spaces has the same objects as theirs but the class of morphisms that we consider strictly includes their morphisms. As a consequence of our duality, the duality of David and Erne easily follows. Using the dual spaces of the mo-distributive posets we prove the existence of a particular Δ1-completion for mo-distributive posets that might be different from the canonical extension. This allows us to show that the canonical extension of a distributive meet-semilattice is a completely distributive algebraic lattice.

Tensor products of semilattices and fuzzy ideals

We study (complete) fuzzy ideals on semilattices from the point of view of tensor products of semilattices. We show that the lattice of all complete fuzzy ideals on a semilattice is an extension of tensor product. We define the notion of semicomplete bi-ideals, and show that the lattice of all fuzzy ideals on a distributive lattice is isomorphic to the lattice of all semicomplete bi-ideals. Moreover, we show that the lattice of all (complete) fuzzy ideals on a distributive (algebraic) lattice is distributive.

Prime algebraicity

A prime algebraic lattice can be characterised as isomorphic to the downwards-closed subsets, ordered by inclusion, of its complete primes. It is easily seen that the downwards-closed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion. The converse also holds; any completely distributive algebraic lattice is isomorphic to such a set of downwards-closed subsets of a partial order. The partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes. Consequently prime algebraic lattices are precisely the completely distributive algebraic lattices. The result extends to Scott domains. Several consequences are explored briefly: the representation of Berry’s dI-domains by event structures; a simplified form of information systems for completely distributive Scott domains; and a simple domain theory for concurrency.

Non-representable distributive semilattices

We present two examples of distributive algebraic lattices which are not isomorphic to the congruence lattice of any lattice. The first such example was discovered by F. Wehrung in 2005. One of our examples is defined topologically, the other one involves majority algebras. In particular, we prove that the congruence lattice of the free majority algebra on (at least) ℵ 2 generators is not isomorphic to the congruence lattice of any lattice. Our method is a generalization of Wehrung’s approach, so that we are able to apply it to a larger class of distributive semilattices.

Distributive congruence lattices of congruence-permutable algebras

We prove that every distributive algebraic lattice with at most ℵ 1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ 1 bound is optimal, as we find a distributive algebraic lattice D with ℵ 2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E.T. Schmidt from 1969. Furthermore, D may be taken as the congruence lattice of the free bounded lattice on ℵ 2 generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilattice-valued ‘distances’ used by B. Jónsson in his proof of Whitman's Embedding Theorem. In particular, the semilattice of compact elements of D is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive 〈 ∨ , 0 〉 -semilattice is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.

Local separation in distributive semilattices

We introduce the Local Separation Property (LSP) for distributive semilattices. We show that LSP holds in many semilattices of the form ConcA, where A is a lattice. On the other hand, we construct an abstract example of a distributive lattice without LSP. Our research is connected with the well known open problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of some lattice.

Lattices of quotients of completely distributive lattices

In this paper, we consider the complete lattice Q(L) of all quotients of a completely distributive lattice L. We show that Q(L) is not a completely distributive lattice even for L a completely distributive algebraic lattice. Some necessary and sufficient conditions for Q(L) to be a completely distributive lattice are given.

Lifting retracted diagrams with respect to projectable functors

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finiteBoolean 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings, can be lifted, with respect to the Concfunctor, by a diagram of lattices, then so can every diagram, indexed by a lattice, of finite distributive 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of our method to other functors, such as the \(R \mapsto V(R)\) functor on von Neumann regular rings.

A uniform refinement property for congruence lattices

The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt's construction or from the approach of P. Pudlak, M. Tischendorf, and J. Tuma. In a previous paper, we constructed a distributive algebraic lattice $A$ with $\aleph_2$ compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice $A$ cannot be obtained using the Pudlak, Tischendorf, Tuma approach. The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, which is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.

On equational theories of semilattices with operators

In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

Standard completions for quasiordered sets

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators: (i) V-distributive completions, (ii) Completely distributive completions, (iii) A-completions (i.e. standard completions which are completely distributive algebraic lattices), (iv) Boolean completions.

A reduced free product of distributive lattices. I

This paper was inspired by the problem whether every distributive algebraic lattice can be represented as the congruence lattice of a lattice. However, here we do not attack this problem directly. Our "conjecture" is that the problem is closely connected with the theory of free distributive lattices and free distributive products. In the present paper we start a systematic study of those questions of the theory of free distributive products that we think relevant in this context. Applications to the representation problem are given in [2]. Let D1 and D~ be finite distributive lattices such that D1 is isomorphic with a 0--V-subsemilattice of D2. This isomorphism will be denoted by 6: a( 6 D1) ~--,-a +.

A representation theorem for completely join distributive algebraic lattices

A representation theorem for completely join distributive algebraic lattices