Pure Compactifications in Quasi-Primal Varieties

1976 ◽  
Vol 28 (1) ◽  
pp. 50-62 ◽  
Author(s):  
Walter Taylor
Keyword(s):  
General Theory ◽  
Representation Theorem ◽  
Boolean Algebras ◽  
Finite Algebras ◽  
Stone Representation

We prove that is quasi-primal, then every algebra in HSPhas a pure embedding into a product of finite algebras. For a general theory of varieties for which every can be purely embedded in an equationally compact algebra , and for all notions not explained here, the reader is referred to [38; 6; or 5]. This theorem was known for Boolean algebras simply as a corollary of the Stone representation theorem and the fact that in the variety of Boolean algebras, all embeddings are pure [2].

Many-Valued Logic and Zadeh’s Fuzzy Sets: A Stone Representation Theorem for Interval-Valued Łukasiewicz–Moisil Algebras

2016 ◽  
Vol 25 (2) ◽  
pp. 99-106
Author(s):  
Abdelaziz Amroune ◽  
Lemnaouar Zedam ◽  
Bijan Davvaz
Keyword(s):  
Representation Theory ◽  
Fuzzy Sets ◽  
Representation Theorem ◽  
Boolean Algebras ◽  
Semantic Interpretation ◽  
Fuzzy Sets Theory ◽  
Logical Basis ◽  
Sets Theory ◽  
Stone Representation ◽  
Interval Valued

AbstractThe aim of this article is to develop a representation theory of interval-valued Łukasiewicz–Moisil algebras; the concept of interval fuzzy sets involves the role that the notion of field of sets plays for the representation of Boolean algebras. This theory provides both a semantic interpretation of a Łukasiewicz interval-valued logic and a logical basis for the interval fuzzy sets theory.

scholarly journals On Fuzzy Ideals ofBL-Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Fuzzy Set ◽  
Representation Theorem ◽  
Prime Ideals ◽  
Converse Implication ◽  
Fuzzy Ideal ◽  
Stone Representation

In this paper we investigate further properties of fuzzy ideals of aBL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of aBL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime idealωis a fuzzy irreducible ideal if and only ifω0=1and|Im⁡(ω)|=2. We give the Krull-Stone representation theorem of fuzzy ideals inBL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of aBL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.

FAST ISOMORPHISM TESTING IN ARITHMETICAL VARIETIES

2003 ◽  
Vol 13 (04) ◽  
pp. 499-506
Author(s):  
TOMASZ A. GORAZD
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Subdirectly Irreducible ◽  
Finite Algebras ◽  
Element Chain ◽  
Subdirectly Irreducible Algebras

Let [Formula: see text] be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from [Formula: see text] have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from [Formula: see text]. This includes the following classical structures in algebra: • Boolean algebras. • Varieties of rings generated by finitely many finite fields. • Varieties of Heyting algebras generated by an n–element chain.

scholarly journals A noncommutative theory of Bade functionals

1991 ◽  
Vol 33 (1) ◽  
pp. 73-81 ◽  
Author(s):  
Don Hadwin ◽  
Mehmet Orhon
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Boolean Algebras ◽  
Representation Space ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Banach Modules ◽  
Key Concepts ◽  
Stone Representation

Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have(i) α (ax) ≥0,(ii) if α (ax) = 0, then ax = 0.

scholarly journals Non-deterministic algebraization of logics by swap structures1

2018 ◽  
Vol 28 (5) ◽  
pp. 1021-1059 ◽  
Author(s):  
Marcelo E Coniglio ◽  
Aldo Figallo-Orellano ◽  
Ana Claudia Golzio
Keyword(s):  
Computer Science ◽  
Representation Theorem ◽  
Algebraic Logic ◽  
Decomposition Theorem ◽  
Boolean Algebras ◽  
Abstract Algebraic Logic ◽  
Algebraic Models ◽  
Algebraizable Logics ◽  
Semantic Framework ◽  
Finite Matrix

Abstract Multialgebras (or hyperalgebras or non-deterministic algebras) have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency (or LFIs) that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics (as independently introduced by M. Fidel and D. Vakarelov). This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI (which is closely connected with Kalman’s functor), suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics.

scholarly journals Gelfand theorem implies Stone representation theorem of Boolean rings

1995 ◽  
Vol 18 (4) ◽  
pp. 701-704
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Boolean Rings ◽  
Stone Representation ◽  
Stone Theorem

Stone Theorem about representing a Boolean algebra in terms of open-closed subsets of a topological space is a consequence of the Gelfand Theorem about representing aB∗- algebra as the algebra of continuous functions on a compact Hausdorff space.

scholarly journals Representation of Boolean hypolattices

1981 ◽  
Vol 24 (3) ◽  
pp. 389-404 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Representation Theorem ◽  
Boolean Lattice ◽  
Principal Result ◽  
Direct Sum ◽  
Boolean Lattices ◽  
Stone Representation ◽  
Relative Convexity ◽  
Small Product

The principal result is a representation theorem for relatively-distributive, relatively complemented hypolattices with zero, generalizing the Stone representation theorem for a Boolean lattice. It uses the small product of a family of Boolean lattices which are maximal sublattices of the hypolattice. The paper also characterizes the maximal sublattices when the hypolattice is coherent; and it gives several examples of hypolattices, including hypolattices of subgroups and of ideals by direct sum, and examples from relative convexity, relative closure, and cofinality.

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