Integral representation of the strongly closed algebra generated by a Boolean algebra of projections

1999 ◽  
pp. 67-90
Author(s):  
Werner Ricker
Keyword(s):  
Boolean Algebra ◽  
Integral Representation ◽  
Closed Algebra

Pseudocomplemented and Implicative Semilattices

1982 ◽  
Vol 34 (2) ◽  
pp. 423-437 ◽  
Author(s):  
C. S. Hoo
Keyword(s):  
Boolean Algebra ◽  
Closed Algebra ◽  
Implicative Semilattices

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.

Wallman Compactification and Representation

1981 ◽  
Vol 33 (2) ◽  
pp. 372-380
Author(s):  
Shankar Hegde
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Lattice Isomorphism ◽  
Linear Map ◽  
Standard Representation ◽  
Closed Algebra ◽  
Image Position ◽  
Uniformly Closed

Let X be any set and A be a uniformly closed algebra of bounded real valued functions on X which contains the constants and separates the points. For a lattice ℒ of subsets of X (we assume throughout that ∅ and X belong to ℒ), let MR(ℒ) denote the space of all finite, finitely additive,ℒ-regular measures defined on the field of sets generated by ℒ . Generalizing the notion of an integral representation, in [5] Kirk and Crenshaw define a standard representation of A*, the Banach dual of A, in MR(ℒ) to be a linear map I of A* into MR(ℒ) with the property that if 0 ≦ ϕ ∈ A*, thenfor every W in ℒ. The space MR(ℒ) is said to represent A* if there exists a (unique) standard representation I of A* onto MR(ℒ) which is a Banach lattice isomorphism.

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 Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

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