scholarly journals Topological representation for monadic implication algebras

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Manuel Abad ◽  
Cecilia Cimadamore ◽  
José Díaz Varela
Keyword(s):  
Boolean Algebra ◽  
Topological Representation ◽  
Implication Algebras ◽  
Implication Algebra ◽  
Monadic Boolean Algebra ◽  
Dual Equivalence ◽  
Implication Spaces ◽  
Monadic Filters

AbstractIn this paper, every monadic implication algebra is represented as a union of a unique family of monadic filters of a suitable monadic Boolean algebra. Inspired by this representation, we introduce the notion of a monadic implication space, we give a topological representation for monadic implication algebras and we prove a dual equivalence between the category of monadic implication algebras and the category of monadic implication spaces.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

scholarly journals Vague Congruences and Quotient Lattice Implication Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoyan Qin ◽  
Yi Liu ◽  
Yang Xu
Keyword(s):  
Lattice Implication Algebra ◽  
Implication Algebras ◽  
Homomorphism Theorem ◽  
Similarity Relations ◽  
Implication Algebra ◽  
Congruence Theory ◽  
Congruence Relations

The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given.

scholarly journals Q-Filters of Quantum B-Algebras and Basic Implication Algebras

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 573 ◽  
Author(s):  
Xiaohong Zhang ◽  
Rajab Borzooei ◽  
Young Jun
Keyword(s):  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
Quantum Logics ◽  
Filter Theory ◽  
Implication Algebras ◽  
Implication Algebra

The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established.

Algebraic functions in Łukasiewicz implication algebras

2016 ◽  
Vol 26 (02) ◽  
pp. 223-247 ◽  
Author(s):  
Miguel Campercholi ◽  
Diego Castaño ◽  
José Patricio Díaz Varela
Keyword(s):  
Main Tool ◽  
Algebraic Functions ◽  
Implication Algebras ◽  
Implication Algebra ◽  
Tarski Algebras ◽  
Łukasiewicz Implication Algebras ◽  
Finitely Generated Variety ◽  
Element Chain ◽  
Global Representation ◽  
Partial Description

In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.

Topological representation for implication algebras

2004 ◽  
Vol 52 (1) ◽  
pp. 39-48 ◽  
Author(s):  
Manuel Abad ◽  
J. Patricio D�az Varela ◽  
Antoni Torrens
Keyword(s):  
Topological Representation ◽  
Implication Algebras

scholarly journals Fantastic filters of lattice implication algebras

2000 ◽  
Vol 24 (4) ◽  
pp. 277-281 ◽  
Author(s):  
Young Bae Jun
Keyword(s):  
Extension Property ◽  
Equivalent Condition ◽  
Lattice Implication Algebra ◽  
Implication Algebras ◽  
Implication Algebra ◽  
Fantastic Filter ◽  
Positive Implicative Filter

The notion of a fantastic filter in a lattice implication algebra is introduced, and the relations among filter, positive implicative filter, and fantastic filter are given. We investigate an equivalent condition for a filter to be fantastic, and state an extension property for fantastic filter.

scholarly journals Quantum logic as an implication algebra

1970 ◽  
Vol 2 (1) ◽  
pp. 101-106 ◽  
Author(s):  
P.D. Finch
Keyword(s):  
Boolean Algebra ◽  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Unary Operation ◽  
Material Implication ◽  
Logical Implication ◽  
Partially Ordered ◽  
Implication Algebra

For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by “≤”, and which, as a poset, is orthocomplemented by a unary operation of negation. The negation of the proposition x is denoted by NX and the least element in the logic is denoted by 0, we write NO = 1.A binary operation “→” is introduced into a logic, the operation is interpreted as material implication so that “x → y” is a proposition of the logic and is read as “x materially implies y”. If material implication has the properties11. (x → 0) = NX, 12. if x ≤ y then (z → x) ≤ (z → y), 13. if x ≤ y then x ≤ (y ≤ z)= x → z, 14. x ≤ {y → N(y → Nx)}, then the logic is an orthomodular lattice. The lattice operations of join and meet are given by x ∨ y = Nx → N(Nx → Ny) x ∧ y = N(X → N(x → y)) and, in terms of the lattice operations, the material implication is given by (x → y) = (y ∧ x) ∨ NX.Moreover the logic is a Boolean algebra if, and only if, in addition to the properties above, material implication satifies 15. (x → y) = (Ny → Nx).

scholarly journals Hilbert Implication Algebra and Some Properties

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Dejen Gerima Tefera
Keyword(s):  
Comparison Theorem ◽  
Implication Algebras ◽  
Implication Algebra

The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.

Synthesis of Antennas on the Base of the Boolean Algebra Transformations and the Atomic Functions Theory

2005 ◽  
Vol 64 (9) ◽  
pp. 699-712
Author(s):  
Victor Filippovich Kravchenko ◽  
Miklhail Alekseevich Basarab
Keyword(s):  
Boolean Algebra

An Outline of a Theory of Propositional Vagueness

2018 ◽  
Author(s):  
Andrew Bacon
Keyword(s):  
Boolean Algebra ◽  
Total Evidence ◽  
Coarse Grained ◽  
Atomic Boolean Algebra

This chapter presents a series questions in the philosophy of vagueness that will constitute the primary subjects of this book. The stance this book takes on these questions is outlined, and some preliminary ramifications are explored. These include the idea that (i) propositional vagueness is more fundamental than linguistic vagueness; (ii) propositions are not themselves sentence-like; they are coarse grained, and form a complete atomic Boolean algebra; (iii) vague propositions are, moreover, not simply linguistic constructions either such as sets of world-precisification pairs; and (iv) propositional vagueness is to be understood by its role in thought. Specific theses relating to the last idea include the thesis that one’s total evidence can be vague, and that there are vague propositions occupying every evidential role, that disagreements about the vague ultimately boil down to disagreements in the precise, and that one should not care intrinsically about vague matters.

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