Normal lattices

Cornish, William H.

doi:10.1017/s1446788700010041

Cited by 100 publications

(95 citation statements)

References 9 publications

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“…It can also be considered as a subvariety of the variety B^ of pseudocomplemented distributive lattices. It is the variety Bj in the chain of subvarieties [10] and B. A. Davey [11], is the natural generalisation of Theorem 1.2, condition (2): a finite distributive lattice is a relative B n -lattice if and only if it does not have 2" +1 © 1 as a homomorphic image.…”

Section: Suppose That P Is a Non-empty Finite Series-parallel Poset mentioning

confidence: 94%

Relative Ockham lattices: their order-theoretic and algebraic characterisation

Bordalo¹,

Priestley

1990

Glasgow Math. J.

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1. Introduction. Given a variety si of lattice-ordered algebras, a lattice L is said to be a relative si-lattice if every closed interval [a, b] of L may be given the structure of an algebra in si (in other words, is the reduct of a member of si-not necessarily unique). This paper discusses the characterisation in terms of forbidden substructures of finite relative .stf-lattices. We treat a large class of varieties si of distributive-lattice-ordered algebras. For these varieties, the finite algebras can be described dually in terms of finite ordered sets, so that order-theoretic results and techniques prove valuable.Our study was prompted by a desire to set in a wider context the following characterisations of relative de Morgan lattices and of relative Stone lattices.

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Section: Suppose That P Is a Non-empty Finite Series-parallel Poset mentioning

confidence: 94%

Relative Ockham lattices: their order-theoretic and algebraic characterisation

Bordalo¹,

Priestley

1990

Glasgow Math. J.

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“…If both L and L d are normal (relatively normal), then we say that L is doubly normal (doubly relatively normal). The following is a routine verification by using the results of Cornish (1972 Cignoli (1978) proved that every B-completely normal lattice is isomorphic with the lattice of all global sections of a sheaf of chains over a Boolean space. If L is incompletely normal, then our stalks £f p turn out to be chains (see Theorem 0.4 below) and our sheaf {if, n, X) coincides with that of Cignoli (1978).…”

Section: ) X I-> X Is An Isomorphism Ofl Onto the Lattice T(x ¥) Of mentioning

confidence: 94%

Prime ideal characterization of chain based lattices

1980

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Epstein and Horn, in their paper 'Chain based lattices', characterized P, -lattices, and P 2 -lattices in terms of their prime ideals. But no such prime ideal characterization for P 0 -lattices was given. Our main aim in this paper is to characterize P 0 -lattices in terms of their prime ideals. We also give a necessary and sufficient condition for a P-algebra to be a P 0 -lattice (and hence a /Vlattice). Keywords and phrases : B-normal lattices, P-algebra, chain based lattices, sheaf of bounded distributive lattices, continuity axiom, s^{L) is determined by a subset of L. IntroductionTraczyk (1963) introduced the concept of chain based lattices (P 0 -lattices) as an abstraction of Post algebras. Epstein and Horn (1975) studied chain based lattices in detail and obtained a prime ideal characterization of these lattices in special cases, namely, /Vlattices and P 2 -lattices. But no such prime ideal characterization for P olattices was given. In this paper, we give a prime ideal characterization for P olattices. The main tool used in this paper is that every bounded distributive lattice is isomorphic with the lattice of all global sections of a sheaf of bounded distributive lattices over a Boolean space (See Maddana Swamy (1974) and Subrahmanyam (1978)).Epstein and Horn (1975) (Theorem 7.3) proved that the prime ideals of a P 0 -lattice of order n lie in disjoint maximal chains each with at most n -1 elements and they have shown that the converse is not true even in the case of a P-algebra. In this paper, we prove that a P-algebra L is a P 0 -lattice (and hence a P 2 -lattice) if and only if the prime ideals of L lie in disjoint maximal chains each with at most n -1 elements for some integer n and satisfy the continuity axiom (see Definition 3.3 and Theorem 3.5 use, available at https://www.cambridge.org/core/terms. https://doi

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“…In [8] and [32], the denominations above are reversed. We have adopted the version of these definitions in [19], for the reason discussed by the author on [19, p. 78].…”

Section: Topological Characterization For the Boolean Lifting Propertymentioning

confidence: 99%

Algebraic and topological results on lifting properties in residuated lattices

Georgescu

Cheptea

Mureșan

2015

Fuzzy Sets and Systems

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We define lifting properties for universal algebras, which we study in this general context and then particularize to various such properties in certain classes of algebras. Next we focus on residuated lattices, in which we investigate lifting properties for Boolean and idempotent elements modulo arbitrary, as well as specific kinds of filters. We give topological characterizations to the lifting property for Boolean elements and several properties related to it, many of which we obtain by means of the reticulation.

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Normal lattices

Cited by 100 publications

References 9 publications

Relative Ockham lattices: their order-theoretic and algebraic characterisation

Relative Ockham lattices: their order-theoretic and algebraic characterisation

Prime ideal characterization of chain based lattices

Algebraic and topological results on lifting properties in residuated lattices

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