SOME RAMSEY THEORY IN BOOLEAN ALGEBRA FOR COMPLEXITY CLASSES

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

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

Telecommunications and Radio Engineering ◽

10.1615/telecomradeng.v64.i9.10 ◽

2005 ◽

Vol 64 (9) ◽

pp. 699-712

Author(s):

Victor Filippovich Kravchenko ◽

Miklhail Alekseevich Basarab

Keyword(s):

Boolean Algebra

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An Outline of a Theory of Propositional Vagueness

10.1093/oso/9780198712060.003.0003 ◽

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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Recent results in partition (Ramsey) theory for finite lattices

Discrete Mathematics ◽

10.1016/0012-365x(81)90207-7 ◽

1981 ◽

Vol 35 (1-3) ◽

pp. 185-198 ◽

Cited By ~ 8

Author(s):

Hans Jürgen Prömel ◽

Bernd Voigt

Keyword(s):

Ramsey Theory ◽

Finite Lattices

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Assessing the Risk of Democratic Reversal in the United States: A Reply to Kurt Weyland

Political Science and Politics ◽

10.1017/s1049096521000329 ◽

2021 ◽

pp. 1-6

Author(s):

Matias López ◽

Juan Pablo Luna

Keyword(s):

United States ◽

Comparative Analysis ◽

Boolean Algebra ◽

Qualitative Comparative Analysis ◽

The United States ◽

American Democracy ◽

Institutional Dynamics ◽

The Public ◽

Original Analysis ◽

The Impact

ABSTRACT By replying to Kurt Weyland’s (2020) comparative study of populism, we revisit optimistic perspectives on the health of American democracy in light of existing evidence. Relying on a set-theoretical approach, Weyland concludes that populists succeed in subverting democracy only when institutional weakness and conjunctural misfortune are observed jointly in a polity, thereby conferring on the United States immunity to democratic reversal. We challenge this conclusion on two grounds. First, we argue that the focus on institutional dynamics neglects the impact of the structural conditions in which institutions are embedded, such as inequality, racial cleavages, and changing political attitudes among the public. Second, we claim that endogeneity, coding errors, and the (mis)use of Boolean algebra raise questions about the accuracy of the analysis and its conclusions. Although we are skeptical of crisp-set Qualitative Comparative Analysis as an adequate modeling choice, we replicate the original analysis and find that the paths toward democratic backsliding and continuity are both potentially compatible with the United States.

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Logical Decision Problems and Complexity of Logic Programs

Fundamenta Informaticae ◽

10.3233/fi-1987-10102 ◽

1987 ◽

Vol 10 (1) ◽

pp. 1-33

Author(s):

Egon Börger ◽

Ulrich Löwen

Keyword(s):

Fixed Point ◽

Computational Complexity ◽

Decision Problem ◽

Decision Problems ◽

Complexity Classes ◽

Logic Programs ◽

Finite Structures ◽

Syntactic Restrictions

We survey and give new results on logical characterizations of complexity classes in terms of the computational complexity of decision problems of various classes of logical formulas. There are two main approaches to obtain such results: The first approach yields logical descriptions of complexity classes by semantic restrictions (to e.g. finite structures) together with syntactic enrichment of logic by new expressive means (like e.g. fixed point operators). The second approach characterizes complexity classes by (the decision problem of) classes of formulas determined by purely syntactic restrictions on the formation of formulas.

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Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽

10.3390/axioms10030164 ◽

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.

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Congruence pairs of principal MS-algebras and perfect extensions

Mathematica Slovaca ◽

10.1515/ms-2017-0430 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1275-1288

Author(s):

Abd El-Mohsen Badawy ◽

Miroslav Haviar ◽

Miroslav Ploščica

Keyword(s):

Boolean Algebra ◽

Congruence Lattices ◽

Descriptive Solution ◽

The One ◽

Special Case ◽

Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

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Programs as term transformers

Fundamenta Informaticae ◽

10.3233/fi-1980-3403 ◽

1980 ◽

Vol 3 (4) ◽

pp. 419-431

Author(s):

Stefan Sokołowski

Keyword(s):

Boolean Algebra ◽

Mathematical Study ◽

Partial Correctness

Predicates describing the states of computation may be regarded as functions into the Boolean algebra {false, true} and programs as transformers of those functions. If we do not restrict ourselves to this algebra, we get instead terms describing the states of computation and programs transforming the terms. In many cases this approach turns out to be more natural. This paper is a mathematical study of partial correctness and termination of programs in the language of term transformations.

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Information Retrieval Systems, an Algebraic Approach I1

Fundamenta Informaticae ◽

10.3233/fi-1981-4306 ◽

1981 ◽

Vol 4 (3) ◽

pp. 551-603

Author(s):

Zbigniew Raś

Keyword(s):

Information Retrieval ◽

Boolean Algebra ◽

Partially Ordered Set ◽

Ordered Set ◽

Algebraic Approach ◽

Partially Ordered ◽

Retrieval Systems ◽

Information Retrieval Systems ◽

Present Part ◽

L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

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