On the T-partial order and properties

For a t-norm T   on [0,1][0,1], a partial order ⪯T⪯T was recently defined and studied. In this study, for any fixed c∈(0,1)c∈(0,1), we define the set of incomparable elements according to ⪯T⪯T and this set is deeply investigated. By means of the equivalence relation, defined in [14], it is shown the set [0,1]/∼[0,1]/∼, denoting the set of all equivalence classes on t-norms on [0,1][0,1] is uncountably infinite. Finally, with the help of any t-norm T   on [0,1][0,1], it is obtained that the family (Tλ)λ∈(0,1)(Tλ)λ∈(0,1) of t-norms on [0,1][0,1]. If T   is a divisible t-norm, then it is obtained that ([0,1],⪯Tλ)([0,1],⪯Tλ) is a lattice. Thus, we give an answer to an open problem in [10].