𝔪-Stone Lattices

1972 ◽  
Vol 24 (6) ◽  
pp. 1027-1032 ◽  
Author(s):  
B. A. Davey
Keyword(s):  
Distributive Lattice ◽  
Principal Ideal ◽  
Forgetful Functor ◽  
Left Adjoint ◽  
Bounded Distributive Lattice ◽  
Stone Lattice ◽  
Complemented Lattice

A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.

Frontal operators in distributive lattices with a generalized implication

2015 ◽  
Vol 08 (03) ◽  
pp. 1550039
Author(s):  
Sergio A. Celani ◽  
Hernán J. San Martín
Keyword(s):  
Distributive Lattice ◽  
Unary Operation ◽  
Duke Math ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Bounded Distributive Lattice ◽  
Modal Lattices ◽  
Weak Heyting Algebras ◽  
San Martín

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [J. E. Castro and S. A. Celani, Quasi-modal lattices, Order 21 (2004) 107–129]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras (see [S. A. Celani and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100(1–2) (2012) 91–114]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator (see [B. A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4(1) (1974) 316–322; M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386]). We give a Priestley’s style duality for each one of the new classes of structures considered.

On a common abstraction of de Morgan algebras and Stone algebras

1983 ◽  
Vol 94 (3-4) ◽  
pp. 301-308 ◽  
Author(s):  
T. S. Blyth ◽  
J. C. Varlet
Keyword(s):  
Distributive Lattice ◽  
Subdirectly Irreducible ◽  
Bounded Distributive Lattice ◽  
Subdirect Irreducibility ◽  
De Morgan Algebras ◽  
Simple Equations ◽  
Lattice Of Congruences

SynopsisWe consider a common abstraction of de Morgan algebras and Stone algebras which we call an MS-algebra. The variety of MS-algebras is easily described by adjoining only three simple equations to the axioms for a bounded distributive lattice. We first investigate the elementary properties of these algebras, then we characterise the least congruence which collapses all the elements of an ideal, and those ideals which are congruence kernels. We introduce a congruence which is similar to the Glivenko congruence in a p-algebra and show that the location of this congruence in the lattice of congruences is closely related to the subdirect irreducibility of the algebra. Finally, we give a complete description of the subdirectly irreducible MS-algebras.

scholarly journals Lattices whose ideal lattice is Stone

1983 ◽  
Vol 26 (1) ◽  
pp. 107-112 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Distributive Lattice ◽  
Dual Space ◽  
Sufficient Conditions ◽  
Distributive Lattices ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Topological Duality ◽  
Bounded Distributive Lattice ◽  
Image Position ◽  
The Ideal

An elementary fact about ideal lattices of bounded distributive lattices is that they belong to the equational class ℬω of all distributive p-algebras (distributive lattices with pseudocomplementation). The lattice of equational subclasses of ℬω is known to be a chainof type (ω+l, where ℬ0 is the class of Boolean algebras and ℬ1 is the class of Stone algebras. G. Grätzer in his book [7] asks after a characterisation of those bounded distributive lattices whose ideal lattice belongs to ℬ (n≧1). The answer to the problem for the case n = 0 is well known: the ideal lattice of a bounded lattice L is Boolean if and only if L is a finite Boolean algebra. D. Thomas [10] recently solved the problem for the case n = 1 utilising the order-topological duality theory for bounded distributive lattices and in [5] W. Bowen obtained another proof of Thomas's result via a construction of the dual space of the ideal lattice of a bounded distributive lattice from its dual space. In this paper we give a short, purely algebraic proof of Thomas's result and deduce from it necessary and sufficient conditions for the ideal lattice of a bounded distributive lattice to be a relative Stone algebra.

scholarly journals De Morgan Functions and Free De Morgan Algebras

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Yu. M. Movsisyan ◽  
V. A. Aslanyan ◽  
Alex Manoogian
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Boolean Functions ◽  
Bounded Lattice ◽  
De Morgan Algebra ◽  
Bounded Distributive Lattice ◽  
Monotone Boolean Functions ◽  
Free Generators ◽  
De Morgan Algebras

AbstractIt is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.

FREE TERNARY ALGEBRAS

2000 ◽  
Vol 10 (06) ◽  
pp. 739-749 ◽  
Author(s):  
RAYMOND BALBES
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Free Generator ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Bounded Distributive Lattice ◽  
Ternary Algebra ◽  
Free Generators ◽  
Ternary Algebras ◽  
Necessary And Sufficient

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

scholarly journals Maximal subalgebras of Heyting algebras

1986 ◽  
Vol 29 (3) ◽  
pp. 359-365 ◽  
Author(s):  
M. E. Adams
Keyword(s):  
Distributive Lattice ◽  
Binary Operation ◽  
Heyting Algebra ◽  
Heyting Algebras ◽  
Maximal Subalgebras ◽  
Bounded Distributive Lattice ◽  
Proposition 8

AHeyting algebra is an algebra H;∨,∧ →, 0,1) of type (2,2,2,0,0) for which H;∨,∧,0,1) is a bounded distributive lattice and → is the binary operation of relative pseudocomplementation (i.e., for a,b,c∈H,ac ∧≦birr c≦a→b). Associated with every subalgebra of a Heyting algebra is a separating set. Those corresponding to maximal subalgebras are characterized in Proposition 8 and, subsequently, are used in an investigation of Heyting algebras.

Bezout Domains and Rings with a Distributive Lattice of Right Ideals

1986 ◽  
Vol 38 (2) ◽  
pp. 286-303 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Distributive Lattice ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Integral Domains ◽  
Bezout Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains

It is the purpose of this paper to discuss a construction of right arithmetical (or right D-domains in [5]) domains, i.e., integral domains R for which the lattice of right ideals is distributive (see also [3]). Whereas the commutative rings in this class are precisely the Prüfer domains, not even right and left principal ideal domains are necessarily arithmetical. Among other things we show that a Bezout domain is right arithmetical if and only if all maximal right ideals are two-sided.Any right ideal of a right noetherian, right arithmetical domain is two-sided. This fact makes it possible to describe the semigroup of right ideals in such a ring in a satisfactory way; [3], [5].

scholarly journals Endomorphism monoids of distributive double p-algebras

1979 ◽  
Vol 20 (1) ◽  
pp. 81-86 ◽  
Author(s):  
M. E. Adams ◽  
J. Sichler
Keyword(s):  
Distributive Lattice ◽  
Unary Operation ◽  
Bounded Distributive Lattice ◽  
Endomorphism Monoids

A distributive p-algebra is an algebra 〈L; ∨, ∧, *, 0, 1〉 for which 〈L, ∨, ∧, 0, 1〉 is a bounded distributive lattice and * is a unary operation on L such that a ∧ x = 0 if and only if x ≤ a* (i.e. a pseudocomplementation). A distributive double p-algebra is an algebra 〈L; ∨, ∧, *, +, 0, 1〉 in which the deletion of + gives a distributive p-algebra and the deletion of * gives a dual distributive p-algebra, that is a ∨ (x = 1 if and only if x ≥ a+.

Coherence classes of ideals of normal lattices with applications to C(X)

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Themba Dube
Keyword(s):  
Topological Space ◽  
Distributive Lattice ◽  
Equivalence Classes ◽  
Bounded Distributive Lattice ◽  
Tychonoff Spaces ◽  
Lattice Of Ideals

AbstractGiven a topological space X, Jenkins and McKnight have shown how ideals of the ring C(X) are partitioned into equivalence classes — called coherence classes — defined by declaring ideals to be equivalent if their pure parts are identical. In this paper we consider a similar partitioning of the lattice of ideals of a normal bounded distributive lattice. We then apply results obtained herein to augment some of those of Jenkins and McKnight. In particular, for Tychonoff spaces, new results include the following:(a)all members of any coherence class have the same annihilator(b)every ideal is alone in its coherence class if and only if the space is a P-space.

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