PlanetPhysics/Algebraic Category of LMn Logic Algebras

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


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


\L{}ukasiewicz 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. \L{}ukasiewicz-Moisil () logic algebras were defined axiomatically in 1970, in ref. [1], as n-valued logic algebra representations and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of -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 -valued logic algebras, and major published results, the reader is referred to [2].

Definition of \L{Edit

ukasiewicz--Moisil (LM), n-valued logic algebras}

\rm (reported by G. Moisil in 1941, cited in refs. [4]).

A {\it --valued \L ukasiewicz--Moisil algebra}, ({\it --algebra}) is a structure of the form Failed to parse (unknown function "\phii"): {\displaystyle (L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)} , subject to the following axioms:

  • (L1) is a {\it de Morgan algebra}, that is, a bounded distributive lattice with a decreasing involution satisfying the de Morgan property ;
  • (L2) For each , Failed to parse (unknown function "\phii"): {\displaystyle \phii:L\lra L} is a lattice endomorphism;\footnote{ The Failed to parse (unknown function "\phii"): {\displaystyle \phii} 's are called the Chrysippian endomorphisms of .}
  • (L3) For each , Failed to parse (unknown function "\phii"): {\displaystyle \phii(x)\vee N{\phii(x)}=1} and Failed to parse (unknown function "\phii"): {\displaystyle \phii(x)\wedge N{\phii(x)}=0} ;
  • (L4) For each , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \phii\circ\phi_{j}=\phi_{k}} iff ;
  • (L5) For each , implies Failed to parse (unknown function "\phii"): {\displaystyle \phii\leq\phi_{j}} ;
  • (L6) For each and , Failed to parse (unknown function "\phii"): {\displaystyle \phii(N x)=N\phi_{n-i}(x)} .
  • (L7) Moisil's `determination principle': Failed to parse (unknown function "\orc"): {\displaystyle \left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;.}

\begin{exe}\rm Let . This set can be naturally endowed with an --algebra structure as follows:

  • the bounded lattice operations are those induced by the usual order on rational numbers;
  • for each , ;
  • for each and , Failed to parse (unknown function "\phii"): {\displaystyle \phii(j/(n-1))=0} if and otherwise.

\end{exe} Note that, for , , and there is only one Chrysippian endomorphism of is , which is necessarily restricted by the determination principle to a bijection, thus making a Boolean algebra (if we were also to disregard the redundant bijection ). Hence, the `overloaded' notation , which is used for both the classical Boolean algebra and the two--element --algebra, remains consistent. \begin{exe}\rm Consider a Boolean algebra Failed to parse (unknown function "\v"): {\displaystyle (B,\v,\w,{}^-,0,1)} . Let T(B)\mbox{LM}_n</math>-algebra structure as follows:

  • the lattice operations, as well as and , are defined component--wise from Failed to parse (unknown function "\Ld"): {\displaystyle \Ld} ;
  • for each and one has:\\ Failed to parse (unknown function "\ov"): {\displaystyle N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})} and Failed to parse (unknown function "\phii"): {\displaystyle \phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .}


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  1. 1.0 1.1 1.2 Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz algebras., J. Algebra , 16 : 486-495.
  2. 2.0 2.1 2.2 Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, Axiomathes , 16 (1-2): 123-136.
  3. 3.0 3.1 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology , 39 : 249-258.
  4. Cite error: Invalid <ref> tag; no text was provided for refs named GG-CV70,GG2k6
  5. Georgescu, G. and D. Popescu. 1968, On Algebraic Categories, Revue Roumaine de Math\'ematiques Pures et Appliqu\'ees , 13 : 337-342.