On some questions concerning the axiomatisation of WNM-algebras and their subvarieties

The logic of the strongest and the weakest t-norms

“…In this paper we focus on the following axiomatic extensions of MTL (EMTL will be introduced in Section 3): C 3 MTL, G, DP, GHP, WNM, RDP [7,2,14,1,19,31]. The first five are axiomatized as MTL plus, respectively, the following axioms:…”

“…Concerning DP, we have that: [29,2].) Every DP-chain A with more than two elements has a coatom c. Moreover, the operations * and ⇒ of a such chain are the following ones, for every x, y ∈ A:…”

“…Many of the logics mentioned in the preceding sections enjoy the standard completeness: [23,14,31,1,12,7].) Let L be one of MTL, C 3 MTL, G, GHP, WNM, RDP.…”

“…However, we have a weaker notion of standard completeness: as we will see in a moment, this is an approximation of standard completeness. [2].) For c ∈ (0, 1), let [0, 1] c be the DP-chain whose carrier is [0, c] ∪{1}.…”

“…In [29,21] the logic S 3 MTL was introduced, and in [2] it was renamed in DP. The reason for the new name is that, even if DP is not t-norm based, the monoidal operation * of every DP-chain is such that x * y = 0, if x, y < 1, and x * y = min{x, y} otherwise, as it happens for * D .…”