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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)
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