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Goldie extending elements in modular lattices

Shriram K. Nimbhorkar, Rupal C. Shroff (2017)

Mathematica Bohemica

The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element a of a lattice L with 0 is said to be a Goldie extending element if and only if for every b ≤ a there exists a direct summand c of a such that b ∧ c is essential in both b and c . Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations of a decomposition...

Ideals in distributive posets

Cyndyma Batueva, Marina Semenova (2011)

Open Mathematics

We prove that any ideal in a distributive (relative to a certain completion) poset is an intersection of prime ideals. Besides that, we give a characterization of n-normal meet semilattices with zero, thus generalizing a known result for lattices with zero.

Indexed annihilators in lattices

Ivan Chajda (1995)

Archivum Mathematicum

The concept of annihilator in lattice was introduced by M. Mandelker. Although annihilators have some properties common with ideals, the set of all annihilators in L need not be a lattice. We give the concept of indexed annihilator which generalizes it and we show the basic properties of the lattice of indexed annihilators. Moreover, distributive and modular lattices can be characterized by using of indexed annihilators.

Join-semilattices with two-dimensional congruence amalgamation

Friedrich Wehrung (2002)

Colloquium Mathematicae

We say that a ⟨∨,0⟩-semilattice S is conditionally co-Brouwerian if (1) for all nonempty subsets X and Y of S such that X ≤ Y (i.e. x ≤ y for all ⟨x,y⟩ ∈ X × Y), there exists z ∈ S such that X ≤ z ≤ Y, and (2) for every subset Z of S and all a, b ∈ S, if a ≤ b ∨ z for all z ∈ Z, then there exists c ∈ S such that a ≤ b ∨ c and c ≤ Z. By restricting this definition to subsets X, Y, and Z of less than κ elements, for an infinite cardinal κ, we obtain the definition of a conditionally κ-co-Brouwerian...

Modularity and distributivity of the lattice of Σ -closed subsets of an algebraic structure

Ivan Chajda, Petr Emanovský (1995)

Mathematica Bohemica

Let 𝒜 = ( A , F , R ) be an algebraic structure of type τ and Σ a set of open formulas of the first order language L ( τ ) . The set C Σ ( 𝒜 ) of all subsets of A closed under Σ forms the so called lattice of Σ -closed subsets of 𝒜 . We prove various sufficient conditions under which the lattice C Σ ( 𝒜 ) is modular or distributive.

Natural extension of a congruence of a lattice to its lattice of convex sublattices

S. Parameshwara Bhatta, H. S. Ramananda (2011)

Archivum Mathematicum

Let L be a lattice. In this paper, corresponding to a given congruence relation Θ of L , a congruence relation Ψ Θ on C S ( L ) is defined and it is proved that 1. C S ( L / Θ ) is isomorphic to C S ( L ) / Ψ Θ ; 2. L / Θ and C S ( L ) / Ψ Θ are in the same equational class; 3. if Θ is representable in L , then so is Ψ Θ in C S ( L ) .

Currently displaying 61 – 80 of 171