It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of T-conditionality. In this paper, the T-conditionality for T-power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication IT being T-conditional are obtained. Moreover, the sufficient conditions under which a power based implication IT is T∗-conditional are discussed, this discussions give an ideas to construct a t-norm T∗ such that the power based implication IT is T∗-conditional.
Recently, Baczyński et al. (2017) proposed a new family of implication operators called semicopula based implications, which combines a given a priori fuzzy implication and a semicopula. In this paper, firstly, the relationship between the basic properties of the priori fuzzy implication and the semicopula based implication are analyzed. Secondly, the conditions such that the semicopula based implication is a fuzzy implication are studied, the study is carried out mainly in the case that the semicopula is a special family semicopula and the priori fuzzy implication is a (U,N)-implication. Moreover, the case that the semicopula based implication is 2-increasing (directionally decreasing, respectively) is also considered.