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Lattices of Scott-closed sets

Weng Kin Ho, Dong Sheng Zhao (2009)

Commentationes Mathematicae Universitatis Carolinae

A dcpo P is continuous if and only if the lattice C ( P ) of all Scott-closed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C ( P ) . In this paper, we study the order-theoretic properties of C ( P ) for general dcpo’s P . The main results are: (i) every C ( P ) is C-continuous; (ii) a complete lattice L is isomorphic to C ( P ) for a complete semilattice P if and only if L is weak-stably C-algebraic; (iii) for any two complete semilattices...

Metrizable completely distributive lattices

Zhang De-Xue (1997)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to study the topological properties of the interval topology on a completely distributive lattice. The main result is that a metrizable completely distributive lattice is an ANR if and only if it contains at most finite completely compact elements.

On minimal spectrum of multiplication lattice modules

Sachin Ballal, Vilas Kharat (2019)

Mathematica Bohemica

We study the minimal prime elements of multiplication lattice module M over a C -lattice L . Moreover, we topologize the spectrum π ( M ) of minimal prime elements of M and study several properties of it. The compactness of π ( M ) is characterized in several ways. Also, we investigate the interplay between the topological properties of π ( M ) and algebraic properties of M .

On the lattice of additive hereditary properties of finite graphs

Ján Jakubík (2002)

Discussiones Mathematicae - General Algebra and Applications

In this paper it is proved that the lattice of additive hereditary properties of finite graphs is completely distributive and that it does not satisfy the Jordan-Dedekind condition for infinite chains.

Prime ideals in the lattice of additive induced-hereditary graph properties

Amelie J. Berger, Peter Mihók (2003)

Discussiones Mathematicae Graph Theory

An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups,...

Primes, coprimes and multiplicative elements

Melvin F. Janowitz, Robert C. Powers, Thomas Riedel (1999)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to study conditions under which the restriction of a certain Galois connection on a complete lattice yields an isomorphism from a set of prime elements to a set of coprime elements. An important part of our study involves the set on which the way-below relation is multiplicative.

Radical classes of distributive lattices having the least element

Ján Jakubík (2002)

Mathematica Bohemica

Let 𝒟 be the system of all distributive lattices and let 𝒟 0 be the system of all L ∈ 𝒟 such that L possesses the least element. Further, let 𝒟 1 be the system of all infinitely distributive lattices belonging to 𝒟 0 . In the present paper we investigate the radical classes of the systems 𝒟 , 𝒟 0 and 𝒟 1 .

Split structures.

Rosebrugh, Robert, Wood, R.J. (2004)

Theory and Applications of Categories [electronic only]

The completely distributive lattice of machine invariant sets of infinite words

Aleksandrs Belovs, Jānis Buls (2007)

Discussiones Mathematicae - General Algebra and Applications

We investigate the lattice of machine invariant classes. This is an infinite completely distributive lattice but it is not a Boolean lattice. The length and width of it is c. We show the subword complexity and the growth function create machine invariant classes.

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