In this paper, two construction methods have been proposed for uni-nullnorms on any bounded lattices. The difference between these two construction methods and the difference from the existing construction methods have been demonstrated and supported by an example. Moreover, the relationship between our construction methods and the existing construction methods for uninorms and nullnorms on bounded lattices are investigated. The charactertics of null-uninorms on bounded lattice L are given and a contruction method is presented.
Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation U in the unit interval with the neutral element e∈[0,1]. If operation U is continuous, then e=0 or e=1. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element e∈(0,1), which is continuous in the open unit square may be given in [0,1)2 or (0,1]2 as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].
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.