Boolean Threshold Representations of Finite and Rational Łukasiewicz Truth Values: Rigidity, Canonical Refinements, and Rational Direct Limits

Abstract

Building on known Boolean tuple representations of finite-valued MV-algebras, we study a lower-threshold profile realisation of the finite Łukasiewicz chain L_{n+1} = {0, 1/n, …, 1}. The value k/n is encoded by the monotone Boolean string with k initial ones, whose i-th coordinate records whether k/n ≥ i/n. In these coordinates, complemented reversal represents negation, coordinatewise meet and join represent the lattice operations, and a Boolean threshold convolution represents truncated sum. Our main focus is the order-theoretic and inter-resolution structure of this representation. We record, with self-contained proofs, that n coordinates are necessary for every order embedding of the (n+1)-element chain into a Boolean cube — an instance of the classical 2-dimension of chains — and that every embedding of minimum dimension is obtained from the threshold encoding by permuting coordinates; these facts single out the threshold encoding as the canonical coordinate system for the analysis that follows. We also give a self-contained account of the induced compositional translation into classical threshold formulas. For m dividing n, coordinate repetition defines an injective MV-homomorphism from the profile algebra at resolution m to that at resolution n. We prove that these are the only MV-homomorphisms between profile algebras: none exist when m does not divide n, and the repetition map is unique when m divides n; in particular every profile algebra is rigid. The resulting directed system is therefore canonical, and its direct limit is isomorphic to the rational MV-algebra Q ∩ [0,1]. Finally, for each fixed MV-formula φ, lower quantisation at resolution n approximates its standard [0,1]-semantics uniformly over valuations, with error at most L(φ)/n.