Localizations and Fractions in Algebra of Logic
The theory of localization and (maximal) algebra of fractions originates in ring theory and was extended by the Romanian school to bounded distributive lattices, Hilbert algebras, Heyting algebras, BLalgebras, MV algebras, Łukasiewicz-Moisil algebras and other algebras related to logic.We have noticed that for each class of algebras under consideration the theory begins with a construction specific to that class and continues by following a pattern common to all classes. In this paper we point out the universal-algebra background of this pattern. Our axiomatic approach provides common generalizations of the parallel theorems in the literature and reveals as a by-product that for each class under discussion the hypotheses of certain theorems can be weakened. The extent of our axiomatic framework includes also R-modules and other classes of algebras.
Keywords: Localization algebra, algebra of fractions, multiplier, bounded distributive lattice, Hilbert algebra, Hertz algebra, Heyting algebra, BL algebra, MV algebra, Łukasiewicz-Moisil algebra, pseudo-BL algebra, pseudo-MV algebra, R-module, commutative ring.
MSC Primary: 03G25 Secondary: 06D20, 06D30, 06D99, 08A99, 16D99