Southeast Asian Bulletin of Mathematics 2001) 25: 47±60 Southeast Asian Bulletin of Mathematics : Springer-Verlag 2001 Sums and Limits of Generalized Direct Families of Algebras* Miroslav C  iric  and Tatjana Petkovic  University of Nis Ï, Faculty of Philosophy, Department of Mathematics, 18000 Nis Ï, C  irila i Metodija 2, Yugoslavia E-mail address: mciric@archimed.®lfak.ni.ac.yu E-mail address: tanjapet@archimed.®lfak.ni.ac.yu Stojan Bogdanovic  University of Nis Ï, Faculty of Economics, 18000 Nis Ï, Trg JNA 11, Yugoslavia E-mail address: sbogdan@archimed.®lfak.ni.ac.yu Abstract. In this paper we introduce a generalization of direct families of algebras and we study their limits and sums. In the case of generalized direct families of algebras carried by idempotent algebras we investigate some subdirect decompositions of their sums. The results that we obtain generalize various results given by J.L. Chrislock and T. Tamura [2], M. C  iric  and S. Bogdanovic  [3±7], H. Mitsch [13], M. Petrich [14±16], B.M. Schein [23±24] and others. 1. Introduction and Preliminaries If A is an algebra of type t without nullary operation symbols, then it can be decomposed by congruence relations all classes of which are subalgebras of A, or equivalently, by congruence relations whose related factors are idempotent alge- bras of type t. Such decompositions are called idempotent decompositions. There are two general problems concerning idempotent decompositions. The ®rst one is the decomposition problem: How should an algebra be decomposed into its components, possibly of simpler structure, and the structure of the whole algebra be described in terms of the structure of the components? The opposite one is the composition problem: Given a family fA i g i A I of pairwise disjoint algebras of type t indexed by an idempotent algebra I of the same type, how should oper- ations of type t on A  6 i A I A i be de®ned so that A becomes an algebra of type t, the equivalence relation on A determined by the partition fA i g i A I is a congruence relation and the related factor algebra is isomorphic to I ? *Supported by Grant 04M03B of RFNS through Math. Inst. SANU