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In this paper, we define an equivalence relation using level set of an i-v L-fuzzy ideal of nearing N. We use this equivalence relation to define upper and lower approximation of nonempty subset of the nearing N. We study properties of these approximations. We find relation between approximations in different cases.

Let N be a zero-symmetric (right) nearring with identity. We introduce a partial order in the matrix nearring corresponding to the partial order (defined by Pilz in Near-rings: the theory and its applications, North Holland, Amsterdam, 1983) in N. A positive cone in a matrix nearring is defined and a characterization theorem is obtained. For a convex ideal I in N, we prove that the corresponding ideal $${I^*}$$ I ∗ is convex in $$M_n(N)$$ M n ( N ) , and conversely, if I is convex in $$M_n(N)$$ M n ( N ) , then $${I_{*}}$$ I ∗ is convex in N. Consequently, we establish an order-preserving isomorphism between the p.o. quotient matrix nearrings $$M_n(N)/{I^*}$$ M n ( N ) / I ∗ and $$M_n(N')/{(I')^*}$$ M n ( N ′ ) / ( I ′ ) ∗ where I and $$I'$$ I ′ are the convex ideals of p.o. nearrings N and $$N'$$ N ′ , respectively. Finally, we prove some properties of Archimedean ordering in matrix nearrings corresponding to those in nearrings.

We consider superfluous elements in a bounded lattice with 0 and 1, and introduce various types of graphs associated with these elements. The notions such as superfluous element graph (S(L)), join intersection graph (JI(L)) in a lattice, and in a distributive lattice, superfluous intersection graph (SI(L)) are defined. Dual atoms play an important role to find connections between the lattice-theoretic properties and those of corresponding graph-theoretic properties. Consequently, we derive some important equivalent conditions of graphs involving the cardinality of dual atoms in a lattice. We provide necessary illustrations and investigate properties such as diameter, girth, and cut vertex of these graphs.

In this paper, we dene the notion of fuzzy graph of a nearring N with respect to a level ideal t denoted by (N;;;t ). The primary aim of this notion is to depict graphically the fuzzy character which is concealed algebraically in the examples of 3-prime fuzzy ideals of a nearring N. We nd that if N is a zero-symmetric nearring and is 3prime fuzzy ideal of N, then (N;;;t ) has a special type of symmetry. We call this symmetry as the ideal symmetry of (N;;;t ). We nd the conditions under which the ideal symmetry of (N;;;t ) implies is a 3-prime fuzzy ideal of N. Finally, we obtain a result which nds all the fuzzy cliques of (N;;;t ) whenever is 3-prime and N is zerosymmetric.

We characterize c-prime fuzzy ideals of the polynomial ring (R n [x];+; ) using the notion of ideal symmetry. Mathematics Subject Classication: 16Y30, 03E72, 16Y99

Recent developments in various algebraic structures and the applications of those in different areas play an important role in Science and Technology. One of the best tools to study the non-linear algebraic systems is the theory of Near-rings. The forward note by Gunter Pilz (Johannes Kepler University, Austria) explains about past developments and future prospects in the theory of nearrings and nearfields. Certain applications of nearrings are found in a few chapters. Some of the chapters are independent; however flow is maintained in all the chapters. It also include few chapters of exploratory approach