Abstract. Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called E-connection approach for combining modal logics to an algebraic setting. §1. Introduction. The combination of decision procedures for logical theories arises in many areas of logic in computer science, such as constraint solving, automated deduction, term rewriting, modal logics, and description logics. In general, one has two first-order theories T 1 and T 2 over signatures Σ 1 and Σ 2 , for which validity of a certain type of formulae (e.g., universal, existential positive, etc.) is decidable. These theories are then combined into a new theory T over a combination Σ of the signatures Σ 1 and Σ 2 . The question is whether decidability transfers from T 1 , T 2 to their combination T .One way of combining the theories T 1 , T 2 is to build their union T 1 ∪ T 2 . Both the Nelson-Oppen combination procedure [23,22] and combination procedures for the word problem [26,28,24,7] address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, there are now also solutions for the non-disjoint case [12,31,8,13, 16,4,5], but they always require some additional restrictions since it is easy to see that in the unrestricted case decidability does not transfer. Similar combination problems have also been investigated in modal logic, where one asks whether decidability of (relativized) validity transfers from two modal logics to their fusion [19,29,32,6]. The approaches in [16,4,5] actually generalize these results from equational theories induced by modal logics to more general first-order theories satisfying certain model-theoretic restrictions: the theories T 1 , T 2 must be compatible with their shared theory T 0 , and this shared theory must be locally finite (a condition ensuring that finitely generated models are finite). The theory T i is compatible with the shared theory T 0 iff (i) T 0 ⊆ T i ; (ii) T 0 has a model completion T * 0 ; and (iii) every model of T i embeds into a model of T i ∪ T * 0 .