On strong standard completeness in some MTL Δ 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 [InlineEquation not available: see fulltext.] This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system L∗∞, expanding the language with Δ 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∗∞ is indeed strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.]. Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions. © 2016, Springer-Verlag Berlin Heidelberg.The authors are thankful to an anonymous reviewer for his/her comments that have helped to improve the final layout of this paper. Vidal has been supported by the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) 15-34650L and by the institutional support RVO:67985807. Esteva and Godo have been funded by the FEDER/MINECO Spanish Project TIN2015-71799-C2-1-P and by the Grant 2014SGR-118 from the Catalan Government. Bou thanks the Grant 2014SGR-788 from the Catalan Government.Peer Reviewe