On the Joint Distribution of Observables

This chapter is devoted to algebraic systems of one binary operation. Groups have been used many times before in the text, as permutations, as modules (a commutative group with an exterior ring multiplication), and elsewhere, but here we organize the systems of one binary operation into one chapter and give some additional specialized results. The most general algebraic system we consider in this chapter is the monoid, a set with one associative binary operation. Some special monoids discussed are unitary monoids, cancellative monoids, and groups. We discuss subsystems, quotient systems, morphisms, and the fundamental morphism theorem. In Sections 9.4 and 9.5 we study more specialized results available for groups alone. We study cyclic groups and connect the order of an element with this concept. Several topics such as center, normalizer, conjugacy classes are all organized around inner automorphisms (φ a (x) = a-xa). We then apply these results to answer some questions about elements of prime order. We offer some standard theorems relating direct products and Cartesian products of groups. We define simple groups and solvable groups. We give an inductive proof of the fundamental theory of Abelian groups which is subsumed in Chapter 10 by the direct sum resolution of a finitely generated module.KeywordsCommutative GroupNormal SubgroupConjugacy ClassSimple GroupCyclic GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A Comparison of J. M. Keynes’s Logical Approach to Probability and Any ‘Objective Bayesian’ Approach to Probability Needs to Incorporate All Five Parts of Keynes’s a Treatise on Probability, Not Just Part I

We continue our study of generalized probability from the viewpoint of category theory. Assuming that each generalized probability measure is a morphism, we model basic probabilistic notions within the category cogenerated by its range. It is known that the closed unit interval I = [0, 1], carrying a suitable difference structure, cogenerates the category ID in which the classical and fuzzy probability theories can be modeled. We study generalized probability theories modeled within two different categories cogenerated by a simplex S n = {(x 1, x 2, …, x n) ∈ I n: $$ \mathop \sum \limits_{i = 1}^n $$ x i ≤ 1}, carrying a suitable difference structure; since I and S 1 coincide, for n = 1 we get the fuzzy probability theory as a special case. In the first case, when the morphisms preserve the so-called pure elements, the resulting category S n D, n > 1, and ID are isomorphic and the generalized probability theories modeled in ID and S n D are “the same”. In the second case, when the morphisms map each maximal element to a maximal element, the resulting categories WS n D, n > 1, lead to different models of generalized probability theories. We define basic notions of the corresponding simplex-valued probability theories and mention some applications.

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

Uninorms, as binary operations on the unit interval, have been widely applied in information aggregation.The class of almost equitable uninorms appears when the contradictory information is aggregated.It is proved that among various uninorms of which either underlying t-norm or t-conorm is continuous, only the representable uninorms belong to the class of almost equitable uninorms.As a byproduct, a characterization for the class of representable uninorms is obtained.

Journal Article Preservation theorems for MTL-chains Get access C. J. Van Alten C. J. Van Alten School of Computer Science, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa E-mail: Clint.VanAlten@wits.ac.za Search for other works by this author on: Oxford Academic Google Scholar Logic Journal of the IGPL, Volume 19, Issue 3, June 2011, Pages 490–511, https://doi.org/10.1093/jigpal/jzp088 Published: 29 January 2010 Article history Received: 30 March 2009 Published: 29 January 2010

Uninorms, as binary operations on the unit interval, have been widely applied in fuzzy set theory. In this paper, the class of conjunctive left-continuous uninorms of which the underlying triangular norm is weak nilpotent minimum are discussed. This class of uninorms are characterized via the ordinal sum construction of Clifford for some cases.

On Selected Properties of Uninorm Petri Nets and Their Application in Modeling Knowledge-Based Systems

An illustrative overview is laid out in this paper detailing the different classes of binary aggregation operations on the unit interval [0, 1] with an absorbing element. The authors focused on the characterization of specific well-known classes of these operations, including nullnorms, T-uninorms, S-uninorms and bi-uninorms, along with their interaction.

<p>Wilson admitted in private correspondence in 1923 with Francis Ysidro Edgeworth that he should not have attempted his review of the A Treatise on Probability (1921) in 1923.The reason Wilson gave to Edgeworth in his correspondence was exactly the same as the reason Edgeworth gave Wilson as the reason Edgeworth was writing to Wilson-a lack of understanding about what Keynes was doing in Part II of the A Treatise on Probability. What Keynes had done, of course, was revolutionary. Keynes had built on the earlier, logical theory of Boole .Keynes erected a general theory of probability where numerical probabilities appeared as a special case when the weight of the evidence, w, defined on the unit interval[0,1],where a w=1 defined a complete evidence set upon which to base a decision maker’s estimates of probability, equaled ,approximated or approached 1 .w measured the completeness of the evidence. If the weight of the evidence equaled 1 and the decision maker had linear probability preferences, then precise, exact, numerical probabilities could be calculated.</p><p><br />Keynes ‘s axiom system in Part II of the A Treatise on Probability was set up to deal with both interval valued probabilities ,which are non additive, and numerical(cardinal) probabilities, which are additive .Non additive probabilities are not subject to the mathematical laws of the probability calculus, the addition rule and the multiplication rule. Keynes used interval valued estimates to deal with the problems of uncertainty and ignorance, where missing relevant information, data, knowledge, or evidence meant that a decision maker could not use numerical probabilities because Keynes’s weight of the evidence variable, w, was less than 1.Only in the case of a complete information set could numerical probabilities or a probability distribution be specified.</p><p><br />In 1934, Wilson published a note titled, Boole’s Challenge Problem, in the Journal of the American Statistical Association. It is actually a disguised, camouflaged second review of J M Keynes’s A Treatise on Probability that deals with Keynes’s Part II again. This time, Wilson gets the technical analysis correct right.<br /><br />Wilson’s note, although he himself had no idea about why dealing with conditions of uncertainty and ignorance was relevant when making a decision, totally and completely destroys the logical and intellectual foundations erected by G. L. S.Shackle and Paul Davidson ,as well as Jochen Runde, Shiela Dow,Rod O’Donnell,Tony Lawson,Anna Carabelli,Robert Skidelsky, Gay Meeks,Bradley Bateman,Donald Gillies ,and many other Post Keynesians,Institutionalists,and Keynesian Fundamentalists,who have claimed, on the basis of papers published by Frank Ramsey and Richard Braithwaite, for nearly 40 years that J M Keynes’s theory in the A Treatise on Probability is an ordinal theory that could only be applied some of the time or has to be interpreted as a relative frequency theory to make an sense out of it (Davis, Winslow).</p><p><br />The two books that supposedly claim to deal with Keynes’s approach to uncertainty, The Philosophy of Keynes’ Economics: Probability, Uncertainty, and Convention,2003,edited by Runde and Mizuhara,and Fundamental Uncertainty,2011,edited by Marzetti Dall’Aste Brandolini and Scazzieri contain no mention of either George Boole or interval valued probability,Thus,they have nothing to say about Keynes’s interval valued approach to probability.</p>

An Analysis of Edwin B. Wilson's Secret, Second Review of the 'A Treatise on Probability' in 1934: How It Demonstrated That Keynes's Theory of Probability Was an Interval Valued Approach to Probability and Not an Ordinal Theory

On Keynes's Realization That His Concept of Uncertainty in the General Theory, Defined as an Inverse Function of Weight from the 'A Treatise on Probability', Would Not Have Any Impact at All on Economists Thinking

Representing Spence's integral by elementary functions

Fuzzy points are concepts used in mathematics, particularly in fuzzy set theory and related fields, where the membership of a given set is described using a membership function that allocates values in the unit interval [0, 1] rather than assigning binary values (true or false).This idea of a fuzzy point is used to consider the fuzzification of (ordered) filters in ordered BCI-algebras.Consequently, the concept of fuzzy (ordered) filters is introduced, and several related properties are investigated.The relationships between fuzzy filter, fuzzy (ordered) filter, and fuzzy (ordered) subalgebra are discussed, and characteristics of the fuzzy ordered filter are considered.This paper provides the conditions for a fuzzy set to become a fuzzy-ordered filter.We also investigate when a positive subset forms an ordered filter and examine whether the level and q-sets satisfy the same properties.