On The Rotation Invariance Of T-Norms

In this paper, we deal with the rotation invariance of t-norms. First, we present some basic properties of rotation invariant t-norms. Then we investigate the relationships between the rotation invariance and other properties of t-norms including left continuity, nilpotency, law of contradiction, contrapositive symmetry of IT and Condition (I) put forward by Peng etc. The major result we obtain is that if (T, S, n) is a De Morgan triple, then T satisfies rotation invariance w.r.t n iff n is a strong negation、(T, n) satisfies law of contradiction and (T,S) satisfies Condition (I).Afterwards, we discuss two applications of rotation invariance in fuzzy preference structures, i.e. the study of T-S-semitransitivity indicator and total semi-order indicator under the condition that the t-norms involved are rotation invariant. The relationship between the T-S-semitransitivity indicator of large preference and that of strict preference is investigated and an equivalent description of T-S-semitransitivity indicator is obtained. Then a total semi-order indicator is defined to express the degree to which a preference structure satisfies total semi-order and the relationship between total semi-order indicator and T-S-semitransitivity indicator is.