BL-global representations

Abstract

A subdirect product [Formula: see text] is global if there is a topology on [Formula: see text] which makes of [Formula: see text] the algebra of all global sections of a sheaf whose stalks are the algebras [Formula: see text] with [Formula: see text]. A quasivariety [Formula: see text] has BL-global representations if there is a class [Formula: see text] containing but close to the class of all relatively subdirectly irreducible members of [Formula: see text], satisfying that every member of [Formula: see text] is isomorphic to a global subdirect product whose factors are in [Formula: see text]. The adjective “BL” refers to “Birkhoff-like” since this type of representations are analogous to the classical Birkhoff subdirect representation by relatively subdirectly irreducibles. This paper has two main contributions. The first one is a generalization of a theorem proved in [D. Vaggione, Sheaf representation and Chinese remainder theorems, Algebra Universalis 29 (1992) 232–272] which characterizes the existence of a global representation of an algebra in terms of the solvability of certain congruence systems. The second one is a theorem assuring the existence of BL-global representations for quasivarieties with a near unanimity term.