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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
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##### References:
  Blok, W.; Pigozzi, D., Algebraizable logics, Mem. amer. math. soc., 396, 77, (1989) · Zbl 0664.03042  Burris, S.; Sankappanavar, H.P., A course in universal algebra, (1981), Springer-Verlag New York · Zbl 0478.08001  Chang, C.C., A new proof of the completeness of łukasiewicz axioms, Trans. amer. math. soc., 93, 74-80, (1989) · Zbl 0093.01104  Cignoli, R.; Torrens, A., Free algebras in varieties of BL-algebras with a Boolean retract, Algebra universalis, 48, 55-79, (2002) · Zbl 1058.03077  Cignoli, R.; D’Ottaviano, I.; Mundici, D., Algebraic foundations of many-valued reasoning, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0937.06009  Di Nola, A., Representation and reticulation by quotients of MV-algebras, Ricerche mat., 40, 291-297, (1991) · Zbl 0767.06013  Di Nola, A.; Dvurečenskij, A., State-morphism MV-algebras, Ann. pure appl. logic, 161, 161-173, (2009) · Zbl 1186.06007  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  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  Dvurečenskij, A., Subdirectly irreducible state-morphism BL-algebras, Arch. math. logic, 50, 145-160, (2011) · Zbl 1215.06006  Flaminio, T.; Montagna, F., MV-algebras with internal states and probabilistic fuzzy logics, Internat. J. approx. reason., 50, 138-152, (2009) · Zbl 1185.06007  Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030  Halpern, J.Y., Reasoning about uncertaintity, (2003), MIT Press  Kroupa, T., Every state on a semisimple MV algebra is integral, Fuzzy sets and systems, 157, 2771-2787, (2006) · Zbl 1107.06007  Kroupa, T., Representation and extension of states on MV-algebras, Arch. math. logic, 45, 381-392, (2006) · Zbl 1101.06008  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  Mundici, D., Interpretations of AF C^{&z.star;}-algebras in łukasiewicz sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059  Mundici, D., Averaging the truth value in łukasiewicz logic, Studia logica, 55, 113-127, (1995) · Zbl 0836.03016  Mundici, D., Bookmaking over infinite-valued events, Internat. J. approx. reason., 46, 223-240, (2006) · Zbl 1123.03011  Panti, G., Invariant measures in free MV-algebras, Comm. algebra, 36, 2849-2861, (2008) · Zbl 1154.06008
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