F-multipliers and the Localization of Mv -algebras

The aim of the present paper is to define the localisation of MValgebra of an MV-algebra A with respect to a topology F on A. In the last part of the paper it is proved that the maximal MV-algebra of quotients (defined in [6]) and the MV-algebra of fractions relative to an ∧−closed system (defined in [5]) are MV algebra of localisation. The concept of multiplier for distributive lattices was defined by W. H. Cornish in [9]. J. Schmid used the multipliers in order to give a non–standard construction of the maximal lattice of quotients for a distributive lattice (see [14]). A direct treatment of the lattices of quotients can be found in [15]. In [11], G. Georgescu exhibited the localization lattice LF of a distributive lattice L with respect to a topology F on L in a similar way as for rings (see [13]) or monoids (see [16]). For the case of Hilbert and Heyting algebras, see [1], [2] and respectively [10]. The concepts of MV -algebra of fractions relative to an ∧− closed system of MV -algebra of fractions and of maximal MV -algebra of quotients were defined by the authors ([5], [6]). 1 Definitions and preliminaries Definition 1.1 ([7], [8]) An MV -algebra is an algebra (A,+,∗ , 0) of type (2, 1, 0) satisfying the following equations: (a1) x+ (y + z) = (x+ y) + z, (a2) x+ y = y + x, (a3) x+ 0 = x, (a4) x∗∗ = x,