Algebras of Lukasiewicz’s logic and their semiring reducts. (English) Zbl 1081.06009

Di Nola, A.; Gerla, B.

Litvinov, G. L. (ed.) et al., Idempotent mathematics and mathematical physics. Proceedings of the international workshop, Vienna, Austria, February 3–10, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3538-6/pbk). Contemporary Mathematics 377, 131-144 (2005).

An interesting connection between MV-algebras and semirings is studied.
Let \(A=(A,\oplus,\odot,\neg,0,1)\) be an MV-algebra. Then it is shown that the reducts \((A,\vee,0,\odot,1)\) and \((A,\wedge,1,\oplus,0)\) form lattice-ordered semirings. Moreover, the operation \(\neg\) gives a relation between these reducts showing that they form a coupled semiring. Further, the authors present examples of semimodules which arise naturally from MV-algebras. Finally, possible areas of applications are discussed, in particular the theory of convolution semirings, automata theory, and fuzzy set theory. Thus we may for example consider “many-valued formal languages” or “many-valued automata”. Another very interesting observation shows that we can in fact view Zadeh’s extension principle as a linear bimorphism between semimodules.
For the entire collection see [Zbl 1069.00011].

Reviewer: Rostislav Horcik (Praha)

MSC:

06D35	MV-algebras
16Y60	Semirings
03B50	Many-valued logic
03G20	Logical aspects of Łukasiewicz and Post algebras
68Q70	Algebraic theory of languages and automata
03E72	Theory of fuzzy sets, etc.

Keywords:

MV-algebra; semiring; semimodule; lattice-ordered semiring; many-valued automaton; many-valued formal language; Zadeh’s extension principle