On a superatomic Boolean algebra which is not generated by a well-founded sublattice

2001 ◽  
Vol 123 (1) ◽  
pp. 221-239 ◽  
Author(s):  
Uri Abraham ◽  
Matatyahu Rubin ◽  
Robert Bonnet
Keyword(s):  
Boolean Algebra ◽  
Superatomic Boolean Algebra

A superatomic Boolean algebra with few automorphisms

2001 ◽  
Vol 40 (2) ◽  
pp. 125-129 ◽  
Author(s):  
Matatyahu Rubin ◽  
Sabine Koppelberg
Keyword(s):  
Boolean Algebra ◽  
Superatomic Boolean Algebra

A very thin thick superatomic Boolean algebra

1985 ◽  
Vol 21 (2-3) ◽  
pp. 137-142 ◽  
Author(s):  
Judy Roitman
Keyword(s):  
Boolean Algebra ◽  
Superatomic Boolean Algebra

On essentially low, canonically well-generated Boolean algebras

2002 ◽  
Vol 67 (1) ◽  
pp. 369-396 ◽  
Author(s):  
Robert Bonnet ◽  
Matatyahu Rubin
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Partial Functions ◽  
Superatomic Boolean Algebra ◽  
Complete Set ◽  
Countably Infinite ◽  
Infinite Set

AbstractLet B be a superatomic Boolean algebra (BA). The rank of B (rk(B)). is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B − {0}, then the rank of a in B (rk(a)). is defined to be the rank of the Boolean algebra . The rank of 0B is defined to be −1. An element a ∈ B − {0} is a generalized atom , if the last nonzero cardinal in the cardinal sequence of B ↾ a is 1. Let a, b ∈ . We denote a ˜ b, if rk(a) = rk(b) = rk(a · b). A subset H ⊆ is a complete set of representatives (CSR) for B, if for every a there is a unique h ∈ H such that h ~ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B.Theorem 1. Let B be a Boolean algebra with cardinal sequence . If B is CWG, then every subalgebra of B is CWG.A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1.Theorem 1 follows from Theorem 2.9. which is the main result of this work. For an ESL BA B we define a set FB of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent.(1) Every subalgebra of B is CWG: and(2) FB is bounded.Theorem 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated.

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.

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

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.

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.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

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.

Programs as term transformers

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.

Information Retrieval Systems, an Algebraic Approach I1

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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