In the present paper, we introduce topological notions defined by means of α-open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Lukasiewicz logic in [0, 1]). We introduce T 0 −, T 1 −, T 2 (αHausdorff)-, T 3 (α-regular)and T 4 (αnormal)-separation axioms. Furthermore, the R 0− and R 1− separation axioms are studied and their relations with the T 1 − ...

Theorem A extends to oriented matroids a theorem for graphs due to Stanley [ 131. It contains Zaslavsky’s [ 141 result, published independently the same year, on the number of regions determined by hyperplanes in Rd. Generalizations of Theorem A can be found in [6, 7, 10-121. The number a(M) = t(M; 2,0) is an important invariant of an oriented matroid M. By Theorem A, a(M) counts the number of ...

Let $L$ be an integral and commutative quantale. In this paper, by fuzzifying the notion of generalized neighborhood systems, the notion of $L$-fuzzy generalized neighborhoodsystem is introduced and then a pair of lower and upperapproximation operators based on it are defined and discussed. It is proved that these approximation operators include generalized neighborhood system...

A graph G = (V,E) is called (k, l)-full if G contains a subgraph H = (V, F ) of k|V |− l edges such that, for any non-empty F ′ ⊆ F , |F ′| ≤ k|V (F ′)| − l holds. Here, V (F ) denotes the set of vertices incident to F . It is known that the family of edge sets of (k, l)-full graphs forms a family of matroid, known as the sparsity matroid of G. In this paper, we give a constant-time approximati...

A gain graph is a graph whose oriented edges are labelled invertibly from a group G, the gain group. A gain graph determines a biased graph and therefore has three natural matroids (as shown in Parts I–II): the bias matroid G has connected circuits; the complete lift matroid L0 and its restriction to the edge set, the lift matroid L, have circuits not necessarily connected. We investigate repre...

For any positive integer l we prove that if M is a simple matroid with no (l + 2)-point line as a minor and with sufficiently large rank, then |E(M)| ≤ q r(M)−1 q−1 , where q is the largest prime power less than or equal to l. Equality is attained by projective geometries over GF(q).