Generic substitutions

Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from being a completed task. In manyvalued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these. 1. Preliminaries We work in the context of propositional algebraic logic. A continuous t-norm (tnorm for short) is a continuous function ⋆ from [0, 1] to [0, 1] such that ([0, 1], ⋆, 1) is a commutative monoid for which x ≤ y implies z ⋆ x ≤ z ⋆ y. Each t-norm induces a residuum → via x ≤ (y → z) iff x ⋆ y ≤ z. One checks easily that the usual lattice operations on [0, 1] are term-definable by x ∧ y = x ⋆ (x → y) and x ∨ y = (