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𝒵 -distributive function lattices

Marcel Erné (2013)

Mathematica Bohemica

It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [ X , Y ] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice 𝒪 X are continuous lattices. This result extends to certain classes of 𝒵 -distributive lattices, where 𝒵 is a subset system replacing the system 𝒟 of all directed subsets (for which the 𝒟 -distributive complete lattices are just the continuous...

2-normalization of lattices

Ivan Chajda, W. Cheng, S. L. Wismath (2008)

Czechoslovak Mathematical Journal

Let τ be a type of algebras. A valuation of terms of type τ is a function v assigning to each term t of type τ a value v ( t ) ≥ 0 . For k ≥ 1 , an identity s ≈ t of type τ is said to be k -normal (with respect to valuation v ) if either s = t or both s and t have value ≥ k . Taking k = 1 with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called k -normal (with respect to the valuation v ) if all its identities are k -normal. For any variety V , there is a least...

A binary operation-based representation of a lattice

Mourad Yettou, Abdelaziz Amroune, Lemnaouar Zedam (2019)

Kybernetika

In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new...

A categorical view at generalized concept lattices

Stanislav Krajči (2007)

Kybernetika

We continue in the direction of the ideas from the Zhang’s paper [Z] about a relationship between Chu spaces and Formal Concept Analysis. We modify this categorical point of view at a classical concept lattice to a generalized concept lattice (in the sense of Krajči [K1]): We define generalized Chu spaces and show that they together with (a special type of) their morphisms form a category. Moreover we define corresponding modifications of the image / inverse image operator and show their commutativity...

A characterization of complete atomic Boolean algebra.

Francesc Esteva (1977)

Stochastica

In this note we give a characterization of complete atomic Boolean algebras by means of complete atomic lattices. We find that unicity of the representation of the maximum as union of atoms and Lambda-infinite distributivity law are necessary and sufficient conditions for the lattice to be a complete atomic Boolean algebra.

A characterization of uninorms on bounded lattices via closure and interior operators

Gül Deniz Çayli (2023)

Kybernetika

Uninorms on bounded lattices have been recently a remarkable field of inquiry. In the present study, we introduce two novel construction approaches for uninorms on bounded lattices with a neutral element, where some necessary and sufficient conditions are required. These constructions exploit a t-norm and a closure operator, or a t-conorm and an interior operator on a bounded lattice. Some illustrative examples are also included to help comprehend the newly added classes of uninorms.

A decomposition of homomorphic images of nearlattices

Ivan Chajda, Miroslav Kolařík (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

By a nearlattice is meant a join-semilattice where every principal filter is a lattice with respect to the induced order. The aim of our paper is to show for which nearlattice 𝒮 and its element c the mapping ϕ c ( x ) = 〈 x ∨ c , x ∧ p c 〉 is a (surjective, injective) homomorphism of 𝒮 into [ c ) × ( c ] .

A duality for isotropic median algebras

Miroslav Ploščica (1992)

Commentationes Mathematicae Universitatis Carolinae

We establish categorical dualities between varieties of isotropic median algebras and suitable categories of operational and relational topological structures. We follow a general duality theory of B.A. Davey and H. Werner. The duality results are used to describe free isotropic median algebras. If the number of free generators is less than five, the description is detailed.

A finite word poset.

Erdős, Péter L., Sziklai, Péter, Torney, David C. (2001)

The Electronic Journal of Combinatorics [electronic only]

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