Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for {Ł}ukasiewicz logic and more recent results for Gödel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants $\{ \overline{r} \mid r \in C \}$, for a suitable countable $C \subseteq [0, 1]$, and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of {Ł}ukasiewicz, Gödel and Product components, (ii) the set of truth-constants {\em covers} all the unit interval in the sense that each component of the t-norm contains at least one value of $C$ different from the bounds of the component, and (iii) the truth-constants in {Ł}ukasiewicz components {\em behave} as rational numbers.