algebraic category of LMn -logic algebras

This is a topic entry on the algebraic category of Łukasiewicz–Moisil n-valued logic algebras that provides basic concepts and the background of the modern development in this area of many-valued logics.

0.1 Introduction

The category ℒℳ of Łukasiewicz-Moisil, n-valued logic algebras (LM_n), and LM_n–lattice morphisms, λ_{LM_n}, was introduced in 1970 in ref. [1] as an algebraic category tool for n-valued logic studies. The objects of ℒℳ are the non–commutative LM_n lattices and the morphisms of ℒℳ are the LM_n-lattice morphisms as defined here in the section following a brief historical note.

0.2 History

Łukasiewicz logic algebras were constructed by Grigore Moisil in 1941 to define ‘nuances’ in logics, or many-valued logics, as well as 3-state control logic (electronic) circuits. Łukasiewicz-Moisil (LM_n) logic algebras were defined axiomatically in 1970, in ref. [1], as n-valued logic algebra representations and extensions of the Łukasiewcz (3-valued) logics; then, the universal properties of categories of LM_n -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of LM_n-logic algebras are under consideration as valid candidates for representations of quantum logics, as well as for modeling non-linear biodynamics in genetic ‘nets’ or networks ([3]), and in single-cell organisms, or in tumor growth. For a recent review on n-valued logic algebras, and major published results, the reader is referred to [2].

0.3 Definition of Łukasiewicz–Moisil (LM), n-valued logic algebras

Definition 0.1. (reported by G. Moisil in 1941, cited in refs. [1, 2]).

A n–valued Łukasiewicz–Moisil algebra, (LM_n–algebra) is a structure of the form (L,∨,∧,N, (φi)_{i∈{1,…,n−1}}, 0, 1), subject to the following axioms:

(L1) (L,∨,∧,N, 0, 1) is a de Morgan algebra, that is, a bounded distributive lattice with a decreasing involution N satisfying the de Morgan property N(x ∨ y) = Nx ∧ Ny;
(L2) For each i ∈{1,…,n−1}, φi : L→L is a lattice endomorphism;^∗
(L3) For each i ∈{1,…,n − 1},x ∈ L, φi(x) ∨ Nφi(x) = 1 and φi(x) ∧ Nφi(x) = 0;
(L4) For each i,j ∈{1,…,n − 1}, φi ∘ φ_j = φ_k iff (i + j) = k;
(L5) For each i,j ∈{1,…,n − 1}, i ≤ j implies φi ≤ φ_j;
(L6) For each i ∈{1,…,n − 1} and x ∈ L, φi(Nx) = Nφ_n−i(x).
(L7) Moisil’s ‘determination principle’: $[∀i ∈ {1, ...,n − 1}, φi(x) = φi(y)] implies [x = y ] .$

Example 0.1. Let L_n = {0, 1∕(n − 1),…, (n − 2)∕(n − 1), 1}. This set can be naturally endowed with an LM_n –algebra structure as follows:

the bounded lattice operations are those induced by the usual order on rational numbers;
for each j ∈{0,…,n − 1}, N(j∕(n − 1)) = (n − j)∕(n − 1);
for each i ∈ {1,…,n − 1} and j ∈ {0,…,n − 1}, φi(j∕(n − 1)) = 0 if j < i and = 1 otherwise.

Note that, for n = 2, L_n = {0, 1}, and there is only one Chrysippian endomorphism of L_n is φ₁, which is necessarily restricted by the determination principle to a bijection, thus making L_n a Boolean algebra (if we were also to disregard the redundant bijection φ₁). Hence, the ‘overloaded’ notation L₂, which is used for both the classical Boolean algebra and the two–element LM₂–algebra, remains consistent.

Example 0.2. Consider a Boolean algebra (B,,∧,⁻, 0, 1). Let T(B) = {(x₁,…,x_n) ∈ Bⁿ⁻¹∣x₁ ≤… ≤ x_n−1}. On the set T(B), we define an LM_n-algebra structure as follows:

the lattice operations, as well as 0 and 1, are defined component–wise from L2;
for each (x₁,…,x_n−1) ∈ T(B) and i ∈{1,…,n − 1} one has:
N(x₁,…x_n−1) = (x_n−1,…,x₁) and φi(x₁,…,x_n) = (x_i,…,x_i).

References

[1] Georgescu, G. and C. Vraciu. 1970, On the characterization of centered Łukasiewicz algebras., J. Algebra, 16: 486-495.

[2] Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil Algebras, Axiomathes, 16 (1-2): 123-136.

[3] Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.

[4] Georgescu, G. and D. Popescu. 1968, On Algebraic Categories, Revue Roumaine de Mathématiques Pures et Appliquées, 13: 337-342.

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