Soft Computing

, Volume 21, Issue 7, pp 1643–1652

# Similarity of binary relations based on complete Boolean lattices and related results

Foundations

## Abstract

Rough set theory is concerned with the lower and upper approximations through a binary relation on the universe. Binary relations play an important role in this theory. This paper investigates similarity of binary relations based on complete Boolean lattices. First, rough approximations based on complete Boolean lattices are proposed through the predecessor neighborhood. Second, L-fuzzy topologies induced by binary relations are researched where L is a complete Boolean lattice. Thirdly, similarity of binary relations is introduced by using L-fuzzy topology and the fact that every binary relation is solely similar to some preorder relation is showed. Finally, as its theoretical expansion, a topological problem “when can the given L-fuzzy topology be coincided with the L-fuzzy topology induced by some binary relation where L is a complete Boolean lattice?” is considered and this problem is answered by introducing L-fuzzy approximating spaces.

### Keywords

Binary relation Similarity Complete Boolean lattice Rough approximation L-fuzzy topology L-fuzzy approximating space

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