×

zbMATH — the first resource for mathematics

State morphism MV-algebras. (English) Zbl 1251.06001
A state on an MV-algebra \((A,\oplus,\neg,0)\) is a map \(s : A \to [0,1]\) such that \(s(\neg 0) = 1\) and if \(\neg(\neg x \oplus \neg y) = 0\) then \(s(x \oplus y) = s(x) + s(y)\). Truth valuations, that is, homomorphisms \(A \to [0,1]\), are special states, called extremal, since every state arises as a convex combination of extremal ones. States stand to MV-algebras as probability distributions stand to Boolean algebras. An SMV-algebra \((A,\tau)\) is an MV-algebra \(A\) with an additional unary operation, called an internal state, satisfying conditions analogous to those defining states. Since these conditions are equational, SMV-algebras form a variety. A state morphism MV-algebra, SSMV-algebra for short, is an SMV-algebra \((A,\tau)\) such that \(\tau\) is an MV-endomomorphism, that is, \(\tau(x \oplus y) = \tau(x) \oplus \tau(y)\).
The paper offers several characterisation results for SMV- and SMMV-algebras. In particular, the first result is a complete characterisation of subdirectly irreducible SMV-algebras.
A second result characterises subdirectly irreducible SMMV-algebras partitioning them into three classes, according to some properties of the underlying MV-algebras and of the associated internal state. A fourth type of subdirectly irreducible SMMV-algebras, properly contained in one of the three aforementioned classes, is introduced. The paper next studies subvarieties of SMMV-algebras and their generators, proving in particular that the whole variety is generated by the diagonalisation of \([0,1]\), that is, by \(([0,1] \times [0,1],\tau)\), where \(\tau(a,b) = (a,a)\) for all \(a \in [0,1]\). As a consequence of this result the authors prove decidability for the whole variety of SMMV-algebras.
The paper then moves on to prove that every subdirectly irreducible SMMV-algebra is subdiagonal, that is, it is a subalgebra of \((B \times C, \tau)\), for some pair of MV-chains \(B,C\), and where \(\tau(b,c) = (b,b)\). Finally, the authors provide axiomatisations for some subvarieties of SMMV-algebras and prove that there are uncountably many subvarieties of SMMV-algebras.

MSC:
06D35 MV-algebras
03B50 Many-valued logic
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Blok, W.; Pigozzi, D., Algebraizable logics, Mem. amer. math. soc., 396, 77, (1989) · Zbl 0664.03042
[2] Burris, S.; Sankappanavar, H.P., A course in universal algebra, (1981), Springer-Verlag New York · Zbl 0478.08001
[3] Chang, C.C., A new proof of the completeness of łukasiewicz axioms, Trans. amer. math. soc., 93, 74-80, (1989) · Zbl 0093.01104
[4] Cignoli, R.; Torrens, A., Free algebras in varieties of BL-algebras with a Boolean retract, Algebra universalis, 48, 55-79, (2002) · Zbl 1058.03077
[5] Cignoli, R.; D’Ottaviano, I.; Mundici, D., Algebraic foundations of many-valued reasoning, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0937.06009
[6] Di Nola, A., Representation and reticulation by quotients of MV-algebras, Ricerche mat., 40, 291-297, (1991) · Zbl 0767.06013
[7] Di Nola, A.; Dvurečenskij, A., State-morphism MV-algebras, Ann. pure appl. logic, 161, 161-173, (2009) · Zbl 1186.06007
[8] Di Nola, A.; Dvurečenskij, A.; Lettieri, A., On varieties of MV-algebras with internal states, Internat. J. approx. reason., 51, 680-694, (2010) · Zbl 1213.06006
[9] Di Nola, A.; Dvurečenskij, A.; Lettieri, A., Erratum “state-morphism MV-algebras”, Ann. pure appl. logic, 161, 161-173, (2009), Ann. Pure Appl. Logic 161 (2010) 1605-1607 · Zbl 1186.06007
[10] Dvurečenskij, A., Subdirectly irreducible state-morphism BL-algebras, Arch. math. logic, 50, 145-160, (2011) · Zbl 1215.06006
[11] Flaminio, T.; Montagna, F., MV-algebras with internal states and probabilistic fuzzy logics, Internat. J. approx. reason., 50, 138-152, (2009) · Zbl 1185.06007
[12] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[13] Halpern, J.Y., Reasoning about uncertaintity, (2003), MIT Press
[14] Kroupa, T., Every state on a semisimple MV algebra is integral, Fuzzy sets and systems, 157, 2771-2787, (2006) · Zbl 1107.06007
[15] Kroupa, T., Representation and extension of states on MV-algebras, Arch. math. logic, 45, 381-392, (2006) · Zbl 1101.06008
[16] Kühr, J.; Mundici, D., De Finetti theorem and Borel states in [0,1]-valued algebraic logic, Internat. J. approx. reason., 46, 605-616, (2007) · Zbl 1189.03076
[17] Mundici, D., Interpretations of AF C^{&z.star;}-algebras in łukasiewicz sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059
[18] Mundici, D., Averaging the truth value in łukasiewicz logic, Studia logica, 55, 113-127, (1995) · Zbl 0836.03016
[19] Mundici, D., Bookmaking over infinite-valued events, Internat. J. approx. reason., 46, 223-240, (2006) · Zbl 1123.03011
[20] Panti, G., Invariant measures in free MV-algebras, Comm. algebra, 36, 2849-2861, (2008) · Zbl 1154.06008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.