A note on the structure of PiMTL-chains and left-continuous cancellative T-norms

Recently the question of standard completeness of MTL and its extensions were deeply studied. Jenei and Montagna showed that MTL logic satisfies standard completeness theorem (see [6]). In [2], the authors studied standard completeness of ΠMTL which is an axiomatic extension of MTL logic where the conjunction is interpreted by a left-continuous cancellative t-norm. They showed that ΠMTL logic is standard complete w.r.t. all ΠMTL-chains whose underlying sets are rational numbers from the unit interval, i.e. [0, 1] ∩ Q. However, they were not able to extend this result to the whole unit interval [0, 1]. Thus if we want to solve this problem we have to understand the structure of ΠMTL-chains and also left-continuous cancellative t-norms more deeply. This paper is a step in this direction. It contributes to the study of the structure of such chains. As a byproduct, we also characterize a rel-