## Retractive MV-algebras.(English)Zbl 0857.06006

Given algebras $${\mathbf A}$$ and $${\mathbf B}$$ of the same type, a homomorphism $$\rho: {\mathbf A} \to {\mathbf B}$$ is called retractive provided that there is a homomorphism $$\delta: {\mathbf B} \to {\mathbf A}$$ such that $$\rho\circ \delta= \text{id}_B$$. Note that a retractive homomorphism must be surjective. A congruence relation $$\theta$$ on $${\mathbf A}$$ is called retractive when the canonical projection $$\pi_\theta: {\mathbf A} \to {\mathbf A}/ \theta$$ is retractive. An algebra $${\mathbf A}$$ is retractive provided all its congruence relations are retractive.
In groups, congruence relations can be identified with normal subgroups. A normal subgroup $$N$$ of a group $$G$$ is retractive if and only if $$G$$ splits over $$N$$. Then it follows that $$N$$ is retractive if and only if it is complemented in the lattice of all subgroups of $$G$$. An analogous result holds for Boolean algebras: An ideal $$I$$ of a Boolean Algebra $${\mathbf B}$$ is retractive if and only if the subalgebra generated by $$I$$ is complemented in the lattice of all subalgebras of $${\mathbf B}$$.
This paper is a first approach to the theory of retractive MV-algebras. We consider the relation between retractivity and complementation in the lattice of subalgebras of an MV-algebra. We obtain that an ideal $$I$$ of the MV-algebra $${\mathbf A}$$ is retractive if and only if the subalgebra of $${\mathbf A}$$ generated by $$I$$ is complemented in the lattice of subalgebras of $${\mathbf A}$$. We investigate the retractivity of direct products of MV-algebras and we characterize the finite retractive MV-algebras.

### MSC:

 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
Full Text: