On the Diagram of One Type Modal Operators on Intuitionistic fuzzy sets: Last expanding with $Z_{alpha ,beta }^{omega ,theta

Abstract

Intuitionistic Fuzzy Modal Operator was defined by Atanassov in cite{at3}<br />in 1999. In 2001, cite{at4}, he introduced the generalization of these<br />modal operators. After this study, in 2004, Dencheva cite{dencheva} defined<br />second extension of these operators. In 2006, the third extension of these<br />was defined in cite{at6} by Atanassov. In 2007,cite{gc1}, the author<br />introduced a new operator over Intuitionistic Fuzzy Sets which is a<br />generalization of Atanassov's and Dencheva's operators. At the same year,<br />Atanassov defined an operator which is an extension of all the operators<br />defined until 2007. The diagram of One Type Modal Operators on<br />Intuitionistic Fuzzy Sets was introduced first in 2007 by Atanassov<br />cite{at10}. In 2008, Atanassov defined the most general operator and in<br />2010 the author expanded the diagram of One Type Modal Operators on<br />Intuitionistic Fuzzy Sets with the operator $Z_{alpha ,beta }^{omega }$.<br />Some relationships among these operators were studied by several researchers%<br />cite{at5}-cite{at8} cite{gc1}, cite{gc3}, cite{dencheva}- cite%<br />{narayanan}.<br />The aim of this paper is to expand the diagram of one type modal operators<br />over intuitionistic fuzzy sets . For this purpose, we defined a new modal<br />oparator $Z_{alpha ,beta }^{omega ,theta }$ over intuitionistic fuzzy<br />sets. It is shown that this oparator is the generalization of the operators<br />$Z_{alpha ,beta }^{omega },E_{alpha ,beta },boxplus _{alpha ,beta<br />},boxtimes _{alpha ,beta }.$