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

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].

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.
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