Localization of Bl - Algebras

The aim of the present paper is to define the localization BL-algebra of a BL-algebra A with respect to a topology F on A. In the last part of the paper is proved that the maximal BL-algebra of quotients (defined in [5]) and the BLalgebra of fractions relative to a ∧-closed system (defined in [4]) are BL-algebras of localization. A remarkable construction in ring theory is the localization ring AF associated with a Gabriel topology F on a ring A (for certain issues connected to the therm localization we have in view Chapter IV: Localization in N. Popescu’s book [21]; see also [18] and [22]). In Lambek’s book [16] it is introduced the notion of complete ring of quotients of a commutative ring, as a particular case of localization ring (relative to the topology of dense ideals). Starting from the example of the rings, J. Schimd introduces in [24], [25] the notion of maximal lattice of quotients for a distributive lattice. The central role in this construction is played by the concept of multipliers defined by W. H. Cornish in [9]. Using the model of localization ring, in [12], G. Georgescu defined for a bounded distributive lattice L the localization lattice LF of L with respect to a topology F on L and prove that the maximal lattice of quotients for a distributive lattice is a lattice of localization (relative to the topology of regular ideals). Analogous results we have for lattices of fractions of bounded distributive lattices relative to ∧-closed systems. Received August 22, 2004; revised December 6, 2004. AMS Subject Classification. 06D35, 03G25.