The subalgebra lattice of a Heyting algebra

AbstractThe notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator “⇝” between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that “the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra” is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

Download Full-text

On an interpretation of second order quantification in first order intuitionistic propositional logic

Journal of Symbolic Logic ◽

10.2307/2275175 ◽

1992 ◽

Vol 57 (1) ◽

pp. 33-52 ◽

Cited By ~ 71

Author(s):

Andrew M. Pitts

Keyword(s):

Propositional Logic ◽

Heyting Algebra ◽

Propositional Calculus ◽

Interpolation Theorem ◽

Second Order ◽

Heyting Algebras ◽

Intuitionistic Propositional Logic ◽

First Order ◽

Intuitionistic Propositional Calculus ◽

Algebraic Counterpart

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula ϕ there exists a formula Apϕ (effectively computable from ϕ), containing only variables not equal to p which occur in ϕ, and such that for all formulas ψ not involving p, ⊢ψ → Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

Download Full-text

ON SEMI-MODULAR SUBALGEBRAS OF LIE ALGEBRAS OVER FIELDS OF ARBITRARY CHARACTERISTIC

Asian-European Journal of Mathematics ◽

10.1142/s1793557108000254 ◽

2008 ◽

Vol 01 (02) ◽

pp. 283-294 ◽

Cited By ~ 2

Author(s):

DAVID A. TOWERS

Keyword(s):

Lie Algebras ◽

Extensive Study ◽

Two Dimensional ◽

Perfect Field ◽

Characteristic P ◽

Subalgebra Lattice ◽

Closed Fields ◽

The One ◽

Algebraically Closed Fields ◽

Number Of Authors

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. It is shown that, in certain circumstances, including for all solvable algebras, for all Lie algebras over algebraically closed fields of characteristic p > 0 that have absolute toral rank ≤ 1 or are restricted, and for all Lie algebras having the one-and-a-half generation property, the conditions of modularity and semi-modularity are equivalent, but that the same is not true for all Lie algebras over a perfect field of characteristic three. Semi-modular subalgebras of dimensions one and two are characterised over (perfect, in the case of two-dimensional subalgebras) fields of characteristic different from 2, 3.

Download Full-text

Lower semimodular Lie algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500020496 ◽

1999 ◽

Vol 42 (3) ◽

pp. 521-540 ◽

Cited By ~ 1

Author(s):

V. R. Varea

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Maximal Subalgebras ◽

Modular Element ◽

Subalgebra Lattice ◽

The Relationship

This paper is concerned with the relationship between the properties of the subalgebra lattice ℒ(L) of a Lie algebra L and the structure of L. If the lattice ℒ(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice ℒ(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A ≤ L and B is maximal in A (Lie algebras satisfying this condition are called sχ-algebras). Our aim is to characterize lower semimodular Lie algebras and sχ-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modularmaximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.

Download Full-text

Abstract characterization for the automorphism group acting on the subalgebra lattice

Algebra Universalis ◽

10.1007/bf02483884 ◽

1981 ◽

Vol 12 (1) ◽

pp. 256-257 ◽

Cited By ~ 1

Author(s):

Isidore Fleischer

Keyword(s):

Automorphism Group ◽

Abstract Characterization ◽

Subalgebra Lattice

Download Full-text

AUTOMATED COMPOSITIONAL REASONING OF INTUITIONISTICALLY CLOSED REGULAR PROPERTIES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054109006851 ◽

2009 ◽

Vol 20 (04) ◽

pp. 747-762

Author(s):

YIH-KUEN TSAY ◽

BOW-YAW WANG

Keyword(s):

Learning Algorithms ◽

Heyting Algebra ◽

Regular Languages ◽

Safety Properties ◽

Compositional Reasoning ◽

Proof Rules

Analysis of infinitary safety properties with automated compositional reasoning through learning is discussed in the paper. We consider the class of intuitionistically closed regular languages and show that it forms a Heyting algebra and is finitely approximatable. Subsequently, compositional proof rules can be verified automatically and learning algorithms for finitary regular languages suffice. We also establish an axiom to deduce circular compositional proof rules for the infinitary languages.

Download Full-text