Normal forms and representable functions in Moisil logic
Andrei Sipos
TL;DR
The paper determines which $r$-ary functions on the standard $LM_n$-algebra are Moisil representable by formulas via a disjunctive normal form theorem, providing a canonical representing term and a concrete criterion for representability. It proves a TFAE characterization: $f: L_{n+1}^r \to L_{n+1}$ is Moisil representable iff $f(a_1,...,a_r) \in \{0,1,a_1,...,a_r,1-a_1,...,1-a_r\}$ for all inputs, and constructs a representing term $t$ from $J_i$-terms and a selector $s$, yielding a constructive normal form. This leads to a detailed description of free $LM_n$-algebras: $F_n(r) \cong \prod_A A^{\alpha(r,A)}$, with $\alpha(r,A)$ counting generating $r$-tuples for each subalgebra $A$ of $\mathcal{L}_n$, and a refinement for odd $n$ via subalgebras $A_{k,j}$ and Cignoli's formula. Overall, the work provides a concrete, constructive toolkit for analyzing LM_n-algebras and their free objects.
Abstract
In this note, we determine, by a disjunctive normal form theorem, which functions on the standard $n$-nuanced Łukasiewicz-Moisil algebra are representable by formulas and we show how this result may help in establishing the structure of the free algebras in this class.