The aim of this paper is to point out and correct some errors of two theorems for IFSs in Atanassov (Two theorems for intuitionistic fuzzy sets, Fuzzy Sets and Systems, 2000) and bring forward a new theorem for IFSs. In the end, some elementary, but non-standard equalities between IFSs were formulated.

this paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. it incorporates the most important settings of lattice-valued topology, including poslat topology of s.~e.~rodabaugh, $(l,m)$-fuzzy topology of t.~kubiak and a.~v{s}ostak, and $m$-fuzzy topology on $l$-fuzzy sets of c.~guido. moreover, its respe...

We give a geometrical interpretation of the interval-valued fuzzy set (IVFS). Based on the geometrical background, we propose new distance measures between IVFSs and compare these measures with distance measures proposed by Burillo and Bustince (1996) and Grzegorzewski (2004), respectively. Furthermore, we extend three methods for measuring distances between IVFSs to interval-valued intuitionis...

We provide an extension of the notion of chain-valued frame introduced by Pultr and Rodabaugh in [Category theoretic aspects of chain-valued frames: parts I and II, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558] by relaxing the assumption that L be a complete chain. As a result of this investigation we formulate the category L-Frm of L-frames under the weaker assumption that L is a comp...

In our earlier work we have developed the axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralak’s axiomatic theory of scalar cardinalities of fuzzy sets. The cardinality has been defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞). We have shown that each scalar cardinality of inter...

The soft set theory is a newly mathematical tool to deal with uncertain problems. However, the classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, we introduce the average function and propose the interval-valued intuitionistic hesitant fuzzy soft sets (IVIHFSSs) which are a combination of the interval-valued intuitionistic hesitant fuzzy sets and...

In 1965, Zadeh introduced the concept of fuzzy subsets. Latter many authors defined different directions of fuzzy subsets. Turksen (1986) generalized the concept of fuzzy set in terms of interval-valued fuzzy set (IVFS). Several researchers present a number of results using IVFSs. Using these concept of (IVFS) Pal and Shyamal (2006) introduced interval-valued fuzzy matrices and shown several pr...

In this paper, Ginsberg’s/Fitting’s theory of bilattices is invoked as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs), serving on one hand to clarify the exact nature of the relationship between these two common extensions of fuzzy sets, and on the other hand providing a general and intuitively attractive frame...