The central notion of a rough set is indiscernibility based on equivalence relation. Since equivalence relation shows strong bondage in an equivalence class, it forms a Galois connection and the difference between upper and lower approximations is lost. We here introduce two different equivalence relations, the one for upper approximation, and the other for lower approximation, and construct co...

In this paper, the order, size and degree of the nodes of the isomorphic fuzzy graphs are discussed. Isomorphism between fuzzy graphs is proved to be an equivalence relation. Some properties of self complementary and self weak complementary fuzzy graphs are discussed. Keywords—complementary fuzzy graphs, co-weak isomorphism, equivalence relation, fuzzy relation, weak isomorphism.

We prove that the equivalence of pure states of a separable C*-algebra is either smooth or it continuously reduces [0, 1]N/l2 and it therefore cannot be classified by countable structures. The latter was independently proved by Kerr–Li–Pichot by using different methods. We also give some remarks on a 1967 problem of Dixmier. If E and F are Borel equivalence relations on Polish spaces X and Y , ...

We deene an equivalence relation on-terms called shuue-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The shuue-equivalence classes are shown to divide the classes of-equal terms into smaller ones consisting of terms with similar reduction behaviour. We reene-reduction from a relation on terms to a relation on shuue-equivalence classes...

We define an analog of Voiculescu’s free entropy for n-tuples of unitaries u1, . . . , un in a tracial von Neumann algebra M which normalize a unital subalgebra L[0, 1] = B ⊂ M. Using this quantity, we define the free dimension δ0(u1, . . . , un G B). This number depends on u1, . . . , un only up to orbit equivalence over B. In particular, if R is a measurable equivalence relation on [0, 1] gen...

A connection is established between the semantic theories of concurrency and communication in the works of de Bakker and Zucker, who develop a denotational semantics of concurrency using metric spaces instead of complete partial orders, and Milner, who develops an algebraic semantics of communication based upon observational equivalence between processes. His rigid synchronization trees (RSTs) ...

Let ďc be computable reducibility on ceers. We show that for every computably enumerable equivalence relation (or ceer) R with infinitely many equivalence classes, the index sets ti : Ri ďc Ru (with R non-universal), ti : Ri ěc Ru, and ti : Ri ”c Ru are Σ3 complete, whereas in case R has only finitely many equivalence classes, we have that ti : Ri ďc Ru is Π2 complete, and ti : Ri ěc Ru (with R...

Many compilation problems require computing the control dependence equivalence relation which divides nodes in a control ow graph into equivalence classes such that nodes are in the same class if and only if they have the same set of control dependences. In this paper, we show that this relation can be computed in O(E) time by reducing it to a naturally stated graph problem: in a strongly conne...

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal κ. We show the consistency of E ++,λ++ λ-club , the relation of equivalence modulo the non-stationary ideal restricted to Sλ ++ λ in the space (λ++)λ ++ , being continuously reducible to E ++ λ+-club , the relation of equi...

The relationship between less detailed and more detailed versions of data is one of the major issues in processing geographic information. Fundamental to much work in model-oriented generalization, also called semantic generalization, is the notion of an equivalence relation. Given an equivalence relation on a set, the techniques of rough set theory can be applied to give generalized descriptio...