Approximations in n-ary algebraic systems

Abstract

Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.Keywords Rough set · Lower approximation · Upper approximation · Algebraic system · n-ary semigroup · Fuzzy set A survey of related worksIn this section, we describe the motivation and a survey of related works.The rough set theory was introduced by Pawlak in 1981. And Pawlak is acknowledged to be a "father" of rough sets. Both rough set theory and theory of evidence deal with uncertainty. They are strongly connected, but they are different approach to uncertainty. The main difference between the rough set theory (Pawlak 1982) and Dempster Shafer theory of evidence (Shafer 1976) is that Dempster Shafer theory uses B. Davvaz (B) belief functions as a main tool, while rough set theory makes use of sets lower and upper approximations, for example see Skowron and Grzymala-Busse (1994).Rough set theory, a new mathematical approach to deal with inexact, uncertain or vague knowledge, has recently received wide attention on the research areas in both of the real-life applications and the theory itself. Rough set theory is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. There are at least two methods for the development of this theory, the constructive and axiomatic approaches. In constructive methods, lower and upper approximations are constructed from the primitive notions, such as equivalence relations on a universe Quafafou 2000) and neighborhood systems (Wong and Pin Nie 1994;Yao 1998). In Pawlak rough sets, the equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes which have a nonempty intersection with the set. It is well known that a partition induces an equivalence relation on a set and vice versa. The properties of rough sets can thus be examined via either partition or equivalence classes. Rough sets are a suitable mathematical model of vague concepts, i.e., concepts without sharp boundaries. Rough set theory is emerging as a powerful theory dealing with imperfect data (Pawlak 1991). It is an expendin...

Applications of interval valued fuzzy n-ary polygroups with respect to t-norms (t-conorms)

“…Davvaz in [15] studied approximations in algebraic systems. Ameri and Zahedi studied algebraic hypersystems in [16].…”

Construction of TernaryHv-groups and TernaryP-hyperoperations

“…Recently, the research about n-ary hyperstructurs has been initiated by Davvaz and Vougiouklis who introduced these structure in [11] and studied by Davvaz, Dudek, Leoreanu-Fotea, Mirvakili, Vougiouklis, and others, see [8,12,13,20,21]. n-Ary hypergroups are a generalization of hypergroups in the sense of Marty, [see 4,5].…”

Smooth approximations by continuous choice-functions

“…When both measurement and input are fuzzy, the approximation is the only truth we are dealing with. Various methods, borrowed and adapted from all areas, numeric analysis and algebra to functional analysis and differential equations, are put to work -see for example (Coroianu & all, 2019), (Dawaz, 2008), (Ishibuchi & all, 2006) and (Wang, Li, 2019). But remarkably, most methods used to achieve good approximations are based essentially on the continuum of the real numbers and, as I believe, do not use enough the intimate properties of the rational numbers.…”