A method to compute the transitive closure, a transitive opening and a transitive approximation of a reflexive and symmetric fuzzy relation is given. Other previous methods in literature compute just the transitive closure, some transitive approximations or some transitive openings. The proposed algorithm computes the three different similarities that approximate a proximity for the computation...

The $L$-fuzzy approximation operator associated with an $L$-fuzzy approximation space $(X,R)$ turns out to be a saturated $L$-fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$-fuzzy approximation spaces and $L$-(fuzzy) topological spaces.

This paper presents the applications of hierarchical clustering to the generators in a power system. A useful application of fuzzy mathematics is that the correction of clustering results and determination of whether it can obtain correct transitive closure. Thus, the fuzzy transitive closure plays an important role in hierarchical clustering. Based on the fuzzy relation matrix, the hierarchica...

Many database queries, such as reachability and regular path queries, can be reduced to finding the transitive closure of the underlying graph. For calculating the transitive closure of large graphs, a distributed computation framework is required to handle the large data volume (which can approach O(|V |) space). Map Reduce was not originally designed for recursive computations, but recent wor...

The serial transitive closure problem is the problem of, given a directed graph G and a list of edges, called closure edges, which are in the transitive closure of the graph, to generate all the closure edges from edges in G . We give a nearly linear upper bound on the number of steps in optimal solutions to the serial transitive closure problem for the case of graphs which are trees. “Nearly l...

Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considere...

Motivated by both established and new applications, we study navigational query languages for graphs (binary relations). The simplest language has only the two operators union and composition, together with the identity relation. We make more powerful languages by adding any of the following operators: intersection; set difference; projection; coprojection; converse; transitive closure; and the...

This paper presents a fully dynamic algorithm for maintaining the transitive closure of a binary relation. All updates and queries can be computed by constant depth threshold circuits of polynomial size (TC circuits). This places dynamic transitive closure in the dynamic complexity class DynTC, and implies that transitive closure can be maintained in database systems that include first-order up...