Ts-Tribes andTs-Measures

AbstractWe show that any fundamental triangular norm-basedTs-tribe T,s∈(0,∞), is a weakly generated tribe. Consequently, T is aT-tribe for any measurable t-normTif and only if it is aTs-tribe for somes∈(0,∞). Further we prove that eachTs-measurem,s∈(0,∞], defined on aTs-tribe T, is a generated measure; i.e., the irreducible part in the Butnariu–Klement decomposition ofTs-measures is always trivial

A Novel Form of Contextuality Predicting Probabilistic Equivalence between Two Sets of Three Mutually Noncommuting Observables

A novel contextual quantum system of observables is introduced, which predicts the state-independent equality of occurrence probabilities between two sets of triple mutually noncommuting observables.Comment: 18 pages, 8 figures, 1 table, dedicated to the memory of Rene Maye

The Fibonacci sequence in the description of maximal discrete Archimedean t-norms.

There are many arguments for counting with more than two “truth values”; this allows to imitate human reasoning of facts which are not binary. For theoretical reasons, it is natural to use the whole real interval as the scale. However, this brings practical problems: it is difficult, and even impossible, to represent exact values. Often only a small scale of values suffices to express what we need. Therefore, finite chains are frequently used as domains of fuzzy logical operations. Their representation and manipulation are easy. In this paper, we focus on triangular norms (t-norms). The choice of a finite domain admits some operations (Gödel, Łukasiewicz), while it excludes others (all strict ones, including the product). A disadvantage of the Gödel (minimum) t-norm is that repetition of arguments does not change their meaning. This is often desirable to emphasize the statement. (“Words, words, words!”) Thus we do not consider the Gödel operations sufficient for representation of all fuzzy logical statements in human reasoning. of the volume of its 5-dimensional domain; in its discrete versions, nonzero results are even more rare. Thus we are interested in Archimedean t-norms with values “as large as possible”, here in the maximal Archimedean t-norms (=those which are not majorized by other Archimedean t-norms). It was shown in previous works that there is an abundance of discrete t-norms; their number grows fast with the number of elements of the underlying chain (no exponential bound seems to be known). There is also an abundance of Archimedean t-norms. In contrast to that, when we counted the number of maximal Archimedean t-norms, it grows asymptotically exponentially with a mild base

Boolean Subalgebras of Orthoalgebras

We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple conditions defining orthodomains and the additional requirement of having enough directions. Excepting pathologies involving maximal Boolean subalgebras of four elements, it is shown that there is an equivalence between the category of orthoalgebras and the category of orthodomains with enough directions with morphisms suitably defined. Furthermore, we develop a representation of orthodomains with enough directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph approach extends the technique of Greechie diagrams and resembles projective geometry. Using such hypergraphs, every orthomodular poset can be represented by a set of points and lines where each line contains exactly three points.Comment: 43 page

States on symmetric logics: extensions

Исследованы состояния на квантовых логиках множеств. Решена одна известная задача

Quantum logics of idempotents of unital rings

Исследованы квантовые логики идемпотенто