scholarly journals L -Fuzzy Prime Spectrums of ADLs

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Natnael Teshale Amare ◽  
Srikanya Gonnabhaktula ◽  
Ch. Santhi Sundar Raj
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Complete Lattice ◽  
Prime Ideals ◽  
Truth Values ◽  
Boolean Rings ◽  
Distributive Law

The notion of an Almost Distributive Lattice (ADL) is a common abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebra and Boolean rings. In this paper, the set of all L -fuzzy prime ideals of an ADL with truth values in a complete lattice L satisfying the infinite meet distributive law is topologized and the resulting space is discussed.

On an algebra of lattice-valued logic

2005 ◽  
Vol 70 (1) ◽  
pp. 282-318
Author(s):  
Lars Hansen
Keyword(s):  
Boolean Algebra ◽  
Complete Lattice ◽  
Algebraic Theory ◽  
Propositional Calculus ◽  
Boolean Lattice ◽  
Finite Chain ◽  
Axiomatic Theory ◽  
Truth Values ◽  
Algebraic Generalization

AbstractThe purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.

scholarly journals L-Fuzzy Congruences and L-Fuzzy Kernel Ideals in Ockham Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Teferi Getachew Alemayehu ◽  
Derso Abeje Engidaw ◽  
Gezahagne Mulat Addis
Keyword(s):  
Complete Lattice ◽  
Ockham Algebra ◽  
Truth Values ◽  
Kernel Ideal ◽  
Congruence Relations ◽  
Fuzzy Ideal ◽  
Fuzzy Congruence ◽  
Distributive Law ◽  
Ockham Algebras ◽  
Equivalent Conditions

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra A , f , whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.

scholarly journals Imbedding Infinitely Distributive Lattices Completely Isomorphically Into Boolean Algebras

1959 ◽  
Vol 15 ◽  
pp. 71-81 ◽  
Author(s):  
Nenosuke Funayama
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Boolean Algebras ◽  
The Other ◽  
Distributive Lattices ◽  
Algebra 1 ◽  
Infinite Distributivity ◽  
Distributive Law ◽  
Finite Sums ◽  
Finite Products

It is well known that any distributive lattice can be imbedded in a Boolean algebra ([1], [2], [4] and others). This imbedding is in general only finitely isomorphic in the sense that the imbedding preserves finite sums (supremums) and finite products (infimums) (but not necessarily infinite ones). Indeed, in order to be able to be imbedded into a Boolean algebra completely isomorphically (i.e. preserving every supremum and infimum) a distributive lattice L must satisfy the infinite distributive law, as the infinite distributivity holds in Boolean algebras. The main purpose of this paper is to prove that the converse is also true, that is, any infinitely distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 6). Since we show, on the other hand, that any relatively complemented distribu tive lattice is infinitely distributive (Theorem 2), Theorem 6 implies that every relatively complemented distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 4).

scholarly journals Constructing some logical algebras from EQ-algebras

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2747-2760
Author(s):  
Rajab Borzooei ◽  
Narges Akhlaghinia ◽  
Xiao Xin ◽  
Mona Kologani
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Algebraic Structure ◽  
Type Theory ◽  
Stone Algebra ◽  
Algebraic Structures ◽  
De Morgan Algebra ◽  
Truth Values ◽  
Mv Algebras

EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets in [18] introduced various kinds of EQ-algebras such as good, residuated, and lattice ordered EQ-algebras. In any logical algebraic structures, by using various kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, by means of fantastic filters of EQ-algebras we construct MV-algebras. Also, we study prelinear EQ-algebras and introduce a new kind of filter and named it prelinear filter. Then, we show that the quotient structure which is introduced by a prelinear filter is a distributive lattice-ordered EQ-algebras and under suitable conditions, is a De Morgan algebra, Stone algebra and Boolean algebra.

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Complete Characterization ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Partially Ordered ◽  
Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

scholarly journals On Generalized Transitive Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Jing Jiang ◽  
Lan Shu ◽  
Xinan Tian
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Canonical Form ◽  
Transitive Closure ◽  
Fuzzy Algebra

Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considered. Some properties of compositions of incline matrices are also given, and a new transitive incline matrix is constructed from given incline matrices. Finally, the issue of the canonical form of a transitive incline matrix is discussed. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.

A localic approach to minimal prime spectra

1988 ◽  
Vol 103 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Sun Shu-Hao
Keyword(s):  
Distributive Lattice ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Minimal Prime

Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)

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