Bernoulli Automorphisms in Many-valued Logic

In classical propositional logic over finitely many variables no automorphism has any stochastic property, because the dual space is a finite discrete set. In this paper we show that the situation for the infinite-valued Lukasiewicz logic is radically different, by exhibiting a family of Bernoulli automorphisms over two variables. As dynamical systems, these are piecewiselinear area-preserving homeomorphisms of the unit square, preserving the denominators of rational points, chaotic in the sense of Devaney, and enjoying the Bernoulli property. 1. Preliminaries Lukasiewicz infinite-valued logic is the most important logic in which the set of truth-values extends the classical {0, 1}. We give an essential description of the system, referring to [9], [6], [5] for more details. A formula is a term in the language L = (→,¬, 1) over propositional variables {xi : i ∈ I}, where I is finite or countable. The set of truth values is the real unit interval [0, 1], which is given the structure of an algebra M = ([0, 1],→,¬, 1) by setting a → b = min(1, 1 − a + b), ¬a = 1− a, 1 = the real number 1. The free algebra FreeI of the equational class generated by M is the algebra of all functions : [0, 1] → [0, 1] that are induced by some formula; the projection functions xi are free generators. Two formulas are equivalent iff they induce the same function, and a formula is a theorem of Lukasiewicz logic iff it induces the function that has value identically 1. We limit ourselves to the case of finitely many variables, and write n ≥ 1 for I. A rational cellular complex over the n-cube is a finite set W of cells (i.e., compact convex polyhedrons) whose union is [0, 1] and such that: (1) every vertex of every cell of W has rational coordinates; (2) if C ∈ W and D is a face of C, then D ∈ W ; (3) every two cells intersect in a common face. By McNaughton’s theorem, the elements of Freen are exactly those continuous functions t : [0, 1] → [0, 1] for which there exists a complex as above and affine linear functions Fj(x̄) = a 1 jx1 + · · ·+ a n j xn + a n+1 j ∈ Z[x1, . . . , xn], in 1-1 correspondence with the n-dimensional cells Cj of the complex, such that t ↾ Cj = Fj for each j. We refer to [8, §5] for the definition of the prime ideal spectrum of Lukasiewicz logic. For the purposes of this paper, we define the dual space of Freen to be the unit cube [0, 1] with the standard topology; by [9, Lemma 8.1] this means restricting