Rational completeness results for prominent propositional fuzzy logics with truth-constants

In this paper we consider expansions of Lukasiewiz, Product, Godel and Nilpotent Minimum logics with truth-constants for an algebra of rational truth-values. We study the semantics for these logics given by chains defined over the rational unit interval and the completeness properties they provide, with a special attention to the completeness with respect to the canonical chain (i.e. the algebra where each truth-constant is interpreted in its corresponding truth-value).