scholarly journals Discriminator varieties of Boolean algebras with residuated operators

1993 ◽  
Vol 28 (1) ◽  
pp. 239-252 ◽  
Author(s):  
Peter Jipsen
Keyword(s):  
Boolean Algebras ◽  
Discriminator Varieties

scholarly journals Boolean like algebras

10.29007/dzwk ◽  
2018 ◽  
Author(s):  
Antonio Ledda ◽  
Tomasz Kowalski ◽  
Francesco Paoli ◽  
Antonino Salibra
Keyword(s):  
General Class ◽  
Central Element ◽  
Boolean Algebras ◽  
Similarity Type ◽  
Discriminator Varieties ◽  
Boolean Product ◽  
Exact Relationship

Using Vaggione's concept of central element in a double pointed algebra, weintroduce the notion of Boolean like variety as a generalisation ofBoolean algebras to an arbitrary similarity type. Appropriately relaxing therequirement that every element be central in any member of the variety, weobtain the more general class of semi-Boolean like varieties, whichstill retain many of the pleasing properties of Boolean algebras. We provethat a double pointed variety is discriminator iff it is semi-Boolean like,idempotent, and 0-regular. This theorem yields a new Maltsev-stylecharacterisation of double pointed discriminator varieties.Moreover, we point out the exact relationship between semi-Boolean-likevarieties and the quasi-discriminator varieties, and we providesemi-Boolean-like algebras with an explicit weak Boolean productrepresentation with directly indecomposable factors. Finally, we discuss idempotentsemi-Boolean-like algebras. We consider a noncommutative generalisation of Boolean algebras and prove - along the lines of similar results available for pointed discriminator varieties or for varieties with a commutative ternary deduction term that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional operations.

Skew Boolean algebras and discriminator varieties

1995 ◽  
Vol 33 (3) ◽  
pp. 387-398 ◽  
Author(s):  
R. J. Bignall ◽  
J. E. Leech
Keyword(s):  
Boolean Algebras ◽  
Discriminator Varieties ◽  
Skew Boolean Algebras

Information storage and retrieval systems with incomplete information I

1979 ◽  
Vol 2 (1) ◽  
pp. 17-41
Author(s):  
Michał Jaegermann
Keyword(s):  
Incomplete Information ◽  
Information Storage And Retrieval ◽  
Boolean Algebras ◽  
Information Storage ◽  
Storage And Retrieval ◽  
Retrieval Systems ◽  
Theory Of Information

In the paper is developed a theory of information storage and retrieval systems which arise in situations when a whole possessed information amounts to a fact that a given document has some feature from properly chosen set. Such systems are described as suitable maps from descriptor algebras into sets of subsets of sets of documents. Since descriptor algebras turn out to be pseudo-Boolean algebras, hence an “inner logic” of our systems is intuitionistic. In the paper is given a construction of systems and are considered theirs properties. We will show also (in Part II) a formalized theory of such systems.

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 Handbook of boolean algebras

1990 ◽  
Vol 84 (1) ◽  
pp. 136
Author(s):  
Gian-Carlo Rota
Keyword(s):  
Boolean Algebras

On Poset Boolean Algebras

Order ◽  
2003 ◽  
Vol 20 (3) ◽  
pp. 265-290 ◽  
Author(s):  
Uri Abraham ◽  
Robert Bonnet ◽  
Wiesław Kubiś ◽  
Matatyahu Rubin
Keyword(s):  
Boolean Algebras

Common extension of Boolean informational systems

1979 ◽  
Vol 2 (1) ◽  
pp. 63-70
Author(s):  
Tadeusz Traczyk
Keyword(s):  
Information Systems ◽  
Boolean Algebras ◽  
Numerical Characterization ◽  
Common Extension

The notion of numerical characterization of Boolean algebras and coproducts are used to define information systems and to develop the theory of such systems.

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