On strong standard completeness in some MTL$_\Delta$ expansions

In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard BL-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm *, find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra $[0, 1]_*$. This system will be an expansion of MTL (Monoidal t-norm based logic). First, we introduce an infinitary axiomatic system $L^\infty_*$, expanding the language with Delta and countably many truth-constants, and with only one infinitary inference rule, that is inspired in Takeuti-Titani density rule. Then we show that $L^\infty_*$ is indeed strongly complete with respect to the standard algebra $[0,1]_*$. Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0,1] satisfy some regularity conditions.