Anti Fuzzy Equivalence Relation on Rings with respect to t-conorm C

In this paper, by using t-conorms, we define the concept of anti fuzzy equivalence relation and anti fuzzy congruence relation on ring R and we investigate some of their basic properties. Also we define fuzzy ideals of ring R under t-conorms and compare this with fuzzy equivalence relation and fuzzy congruence relation on ring R such that we define new introduced ring. Next we investigate this concept under homomorphism of new introduced ring.


Introduction
A ring is a set equipped with two operations (usually referred to as addition and multiplication) that satisfy certain properties: there are additive and multiplicative identities and additive inverses, addition is commutative, and the operations are associative and distributive. The study of rings has its roots in algebraic number theory, via rings that are generalizations and extensions of the integers, as well as algebraic geometry, via rings of polynomials. These kinds of rings can be used to solve a variety of problems in number theory and algebra; one of the earliest such applications was the use of the Gaussian integers by Fermat, to prove his famous two-square theorem. There are many examples of rings in other areas of mathematics as well, including topology and mathematical analysis. In mathematics, an equivalence relation is a binary relation that is 2 at the same time a reflexive relation, a symmetric relation and a transitive relation. A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases congruence relations have an alternative representation as substructures of the structure on which they are defined. E.g. the congruence relations on groups correspond to the normal subgroups. In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Zadeh [30] and Gottwald [4] in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. The concept of a fuzzy relation was first proposed by Zadeh [30]. Subsequently, Goguen [3] and Sanchez [29] studied fuzzy relations in various contexts. In [7] Nemitz discussed fuzzy equivalence relations, fuzzy functions as fuzzy relations, and fuzzy partitions. Murali [6] developed some properties of fuzzy equivalence relations and certain lattice theoretic properties of fuzzy equivalence relations. Samhan [28] characterized the fuzzy congruences generated by fuzzy relations on a semigroup and studied the lattice of fuzzy congruences on a semigroup. T-conorms are a generalization of the usual two-valued logical conjunction, studied by classical logic, for fuzzy logics. Indeed, the classical Boolean conjunction is both commutative and associative. The monotonicity property ensures that the degree of truth of conjunction does not decrease if the truth values of conjuncts increase. The requirement that 1 be an identity element corresponds to the interpretation of 1 as true (and consequently 0 as false). Continuity, which is often required from fuzzy conjunction as well, expresses the idea that, roughly speaking, very small changes in truth values of conjuncts should not macroscopically affect the truth value of their conjunction. T-conorms are also used to construct the intersection of fuzzy sets or as a basis for aggregation operators (see fuzzy set operations). In probabilistic metric spaces, t-conorms are used to generalize triangle inequality of ordinary metric spaces. Individual 3 t-conorms may of course frequently occur in further disciplines of mathematics, since the class contains many familiar functions. The author by using norms, investigated some properties of fuzzy algebraic structures [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. The purpose of this paper is to deal with the anti fuzzy congruence and equivalence relation on rings with respect to t-conorms, homomorphisms and isomorphisms of them. Section 2 contains some basic definitions and preliminary results which will be needed in the sequel. In Section 3, by using t-norms, we introduce the concept of anti fuzzy equivalence relation, anti fuzzy congruence relation and anti fuzzy ideal on rings and we obtain some results of them. Also we define addition and product of them and introduce new rings. In Section 4, we define the kernel of ring homomorphisms and we prove that it will be a congruence relation on rings. Also we show that anti characteristic function of it is an anti fuzzy congruence relation on rings.
Next we obtain ring homomorphism and ring isomorphisms of it.

Preliminaries
This section contains some basic definitions and preliminary results which will be needed in the sequel. Definition 2.1 (see [5]). A ring ⋅ +, , R consists of a nonempty set R and two binary operations + and ⋅ that satisfy the axioms: is an abelian group; (2) ( ) ( ) (2) Z 5 is a commutative ring with no identity.    (6) Einstein sum (compare the velocity-addition formula under special relativity) is a dual to one of the Hamacher t-norms. Note that all t-conorms are bounded by the maximum and the drastic t-conorm:

t-conorms over Anti Fuzzy Equivalence Relation on Rings
on a ring R is an anti fuzzy equivalence relation on R with respect to t-conorm C if the following conditions are satisfied: on a ring R is an anti fuzzy congruence relation on R with respect to t-conorm C if the following conditions are satisfied: We denote the set of all anti fuzzy congruence relations on R by Then for all , , , R z y x ∈ the following assertions hold: ). , Then from (a) and (b) we get , , 1 , Now by (1) and (c) we obtain that Define a fuzzy subset a µ on R as We denote the set of fuzzy subset a µ on R by . µ R Proposition 3.6. Let ).  , We denote the set of all anti fuzzy ideals of R with respect to t-conorm C by (
Proof. Let . , , R c b a ∈ As Propositions 3.15 and 3.16 we get that It is easy to prove that   is an abelian group. Now will be a ring.
will be an anti fuzzy equivalence relation on R under t-conorm C: and then will be an anti fuzzy equivalence relation on R under t-conorm C: (2) (a) Since ( ) 0 Therefore from (a)-(d) we obtain that is a congruence relation on R.