How to tell when a product of two partially ordered spaces has a certain property: General results with application to fuzzy logic

Abstract

Abstract-In this paper, we describe how checking whether a given property F is true for a product A1 × A2 of partially ordered spaces can be reduced to checking several related properties of the original spaces Ai. I. FORMULATION OF THE PROBLEMDegrees of certainty: from {0, 1} to [0, 1] to general partially ordered sets. In the traditional 2-valued logic, every statement is either true or false. Thus, the set of possible truth values consists of two elements: true (1) and false (0).Fuzzy logic takes into account that people have different degrees of certainty in their statements; see, e.g., [2], [10].Traditionally, fuzzy logic uses values from the interval [0, 1] to describe uncertainty. In this interval, the order is total (linear) in the sense that for every two elements a, a More complex sets of possible degrees are also sometimes useful. Not to miss any new options, in this paper, we consider general partially ordered spaces.Need for product operations. Often, two (or more) experts evaluate a statement S. Then, our certainty in S is described by a pair (a 1 , a 2 ), where a i ∈ A i is the i-th expert's degree of certainty. To compare such pairs, we must therefore define a partial order on the set A 1 × A 2 of all such pairs.Two examples of product operations. One example of a partial order on A 1 × A 2 is a Cartesian product:This product corresponds to a cautious approach, when our confidence in S ′ is higher than in S if and only if it is higher for both experts.Another example is a lexicographic product:This product corresponds to the case when we have the absolute confidence in the first expert; then, we only use the opinion of the second expert when, to the first expert, the degrees of certainty are indistinguishable. We can have other product operations in which the relation between the pairs (a 1 , a 2 ) and (a A natural question. Once a product is defined, it is reasonable to ask when the resulting partially ordered set A 1 × A 2 it satisfies a certain property: is it a total order? is it a lattice order? etc. It is desirable to have some criteria that would transform the question about the product space into questions about related properties of component spaces.Some such criteria are known (see, e.g., [13] and references therein). For example:• A Cartesian product is a total order if and only if one of the components is a total order, and the other consists of a single element.• A lexicographic product is a total order if and only if both components are totally ordered.What we do in this paper. In this paper, we provide a general algorithm that reduced the question whether a certain property is satisfied for a product to several properties of component spaces. Applications beyond orders. Our algorithm does not use the fact that the original relations are orders (i.e., transitive antisymmetric relations). Thus, our algorithm is applicable to a general case when we have an arbitrary binary relationequivalence, similarity, etc. Moreover, this algorithm can be

Reconstructing an Open Order from Its Closure, with Applications to Space-Time Physics and to Logic

“…In this application, to each statement, we assign the expert's degree of certainty that this statement is true. A natural partial ordering relation a b describes the fact that we are more certain in b than in a; see, e.g., [20,34,38,39,43,44].…”