Automorphisms of Powers of Linearly Ordered Sets

نویسندگان

Carol L. Walker

Elbert A. Walker

چکیده

Let I = ([0; 1];_;^; 0; 1), where _ and ^ are max and min, respectively. This algebra is the basic building block of fuzzy set theory and logic. Likewise, the algebra I = ([0; 1];_;^; 0; 1), where [0; 1] = f(a; b) : a; b 2 [0; 1], a bg, _, ^ are given coordinate-wise, and 0 and 1 are the bounds on [0; 1], is the relevant algebra for interval-valued fuzzy set theory and logic. A good share of the theory of fuzzy sets is concerned with endowing I with additional structure such as t-norms, t-conorms, and negations other than the usual max and min operations. In addressing the situation for I, a basic problem was deciding on the appropriate denitions [1]. For example, what should a t-norm on I be? In this regard, knowing the automorphisms of I was fundamental for representation theorems for norms, conorms, and negations. The basic theorem is that any automorphism of I is of the form (a; b)! ('(a); '(b)) where ' is an automorphism of I [1]. Our initial motivation was to extend this theorem from I to I for any positive integer n. But more generally, we will replace [0; 1] by a bounded linearly ordered set S, and consider the automorphisms of both S and S. The result for S is the same as for I except for an anomaly for certain combinations of nite S and integers n (Theorem 7).

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Conditions Equivalent to Unit Representations of Ordered Relational Structures.

This paper studies several concepts about subgroups of automorphisms of linearly ordered relational structures. In particular, it focuses on conditions that are equivalent to the translations (automorphisms with no fixed points plus the identity) forming a homogeneous, Archimedean ordered group under the asymptotic order. For the automorphisms of an ordered relational structure, these propertie...

متن کامل

Symmetric Chain Decompositions of Quotients of Chain Products by Wreath Products

Subgroups of the symmetric group Sn act on powers of chains C n by permuting coordinates, and induce automorphisms of the ordered sets C. The quotients defined are candidates for symmetric chain decompositions. We establish this for some families of groups in order to enlarge the collection of subgroups G of the symmetric group Sn for which the quotient Bn/G obtained from the G-orbits on the Bo...

متن کامل

Tripled partially ordered sets

In this paper, we introduce tripled partially ordered sets and monotone functions on tripled partiallyordered sets. Some basic properties on these new dened sets are studied and some examples forclarifying are given.

متن کامل

General definitions for the union and intersection of ordered fuzzy multisets

Since its original formulation, the theory of fuzzy sets has spawned a number of extensions where the role of membership values in the real unit interval $[0, 1]$ is handed over to more complex mathematical entities. Amongst the many existing extensions, two similar ones, the fuzzy multisets and the hesitant fuzzy sets, rely on collections of several distinct values to represent fuzzy membershi...

متن کامل