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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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