On the Relationship of AP, RS, and CEP in Congruence Modular Varieties, II

Let V be a congruence distributive variety, or a congruence modular variety whose free algebra on 2 generators is finite. If V is residually small and has the amalgamation property, then it has the congruence extension property. Several applications are presented. In two previous papers [1] and [2], we considered the following question: if V is a residually small variety with the amalgamation property, must V have the congruence extension property? Our work established the following implications for a congruence modular variety V : (1) If V is 2–finite and has C2, then AP + RS =⇒ R. (2) If V is 4–finite with C2 and R, then AP + RS =⇒ CEP. (The terminology will be explained below.) In this paper we supplement and extend these results. Assuming still that V is congruence modular, we have: (3) AP + RS =⇒ C2. (4) If V has R, then AP + RS =⇒ CEP. Combining these implications, we have that every congruence modular, 2–finite variety satisfies AP + RS =⇒ CEP. Furthermore, every congruence distributive variety (no finiteness assumption) satisfies AP + RS =⇒ CEP. Our universal algebraic notation and terminology are standard. Good references are [4] and [9]. Let V be a variety of algebras. We say that V • has the amalgamation property (AP) if, for all A,B0,B1 ∈ V and all embeddings fi :A → Bi, for i = 0, 1, there is C ∈ V and embeddings gi :Bi → C, i = 0, 1, such that g0 ◦ f0 = g1 ◦ f1, • is residually small (RS) if there is a cardinal κ such that every subdirectly irreducible algebra in V has cardinality less than κ, • has the congruence extension property (CEP) if, for all A ≤ B ∈ V, and congruence α on A, there is ᾱ ∈ ConB such that ᾱ A = α, • is n–finite, for a positive integer n, if every member of V generated by n elements is finite. 1991 Mathematics Subject Classification. Primary 08B10, 08B25; Secondary 03C25, 20E06.. Research partially supported by the Iowa State University Sciences and Humanities Research Institute and National Science Foundation Grants No. DMS-8600300 and DMS-8701643