Semi-divisible t-norms on discrete scales (Q2457039)

The paper elaborates on the conjunctions for semi-divisible discrete triangular norms. Such t-norms are defined as mappings \(T: L_2\to L\) with \(L= \{0, 1/n, 2/n,\dots,1\}\) such that they satisfy the semi-divisibility of the form \[ (a\to_Tb)\to_T b= (b\to_T a)\to_T a, \] where \(\to_T\) is the residual implication adjoint to \(T\) and \(a,b\in\text{Ran}(n_T)\) and \(n_T: L\to L\) with \(n_T(x)= x\to_T 0\). Several properties and examples of semi-divisible discrete t-norms are presented. It is shown that the only semi-divisible t-norm with \(\text{Ran}(n_T)= L\) corresponds to Łukasiewicz t-norm. In the case of the ordinal sum of t-norms, the only important summand is the first one in the corresponding ordinal sum.