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The variety of RDP-algebras forms the algebraic semantics of RDPlogic, the many-valued propositional logic of the revised drastic product left-continuous triangular norm and its residual. We prove a Priestley duality for finite RDP-algebras, and obtain an explicit description of coproducts of finite RDP-algebras. In this light, we give a combinatorial representation of free finitely generated RDP-algebras, which we exploit to construct normal forms, strongest deductive interpolants, and most general unifiers. We prove that RDP-unification is unitary, and that the tautology problem for RDP-logic is coNP-complete.

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The drastic product * D is known to be the smallest t-norm, since x * D y = 0 whenever x, y < 1. This t-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product t-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called S3MTL in [Nog06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the ∆ projection operator and to discriminator varieties. We shall show that the category of finite DPalgebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras. arXiv:1406.7166v1 [math.LO]

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