Free five-valued Nelson Algebras

TL;DR

This paper develops a representation-theoretic framework for five-valued Nelson algebras using prime filters and deductive systems, and applies it to compute free algebras with finitely many generators. The defining NT$_3$ equation $((x\to z)\to y)\to(((y\to x)\to y)\to y)=1$ characterizes the five-valued subclass, and the authors show every nontrivial algebra decomposes via quotients $N/D$ (with $D$ irreducible) into chains $C_i$ for $2\le i\le 5$, enabling a subdirect product representation and a Priestley-type dual description in the finite case. The main result is a closed-form counting formula for the free algebras: $|F(n)| = 2^{2^{2n}} \prod_{k=1}^{n} (2^{2^{n-k}}+1)^{\binom{n}{k} 2^{n-k}}$, with $|F(1)|=48$, confirming Brignole's earlier but unpublished insights and providing a complete structural account of free objects in this variety. The work yields a concrete, modular view of five-valued Nelson algebras as subalgebras of products of small chains, with explicit combinatorial data for free generators.

Abstract

Five-valued Nelson algebras are those satisfying the condition: $((x\to z)\to y)\to(((y \to x)\to y)\to y)=1$. We give alternative equations defining these algebras, and determine the structure and number of elements of the free five-valued Nelson algebra with a finite number of free generators.

Theorems & Definitions (38)

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