Relative MV-algebras and relative homomorphisms

In the paper is defined the notion of relative subalgebra of an MV-algebra A. A particular case of this notion is the notion of an interval subalgebra of A, which has been already studied in the literature. Applying these notions, two new categories denoted as rMV and intMV are introduced. In both cases the objects are MV-algebras, but the homomorphisms are defined by means of relative subalgebras or by interval subalgebras, respectively. The relations occurring between these categories and the category of all MV-algebras with usual homomorphisms are investigated. The main results of the paper deal with one-generated free MV-algebras in the variety generated by the finite MV-chains Si , 0<i< p +1 (p varying over the set of all positive integers) and their relations to certain relative subalgebras of the cyclic free MV-algebra.