6

PETER H. KRAUSS AND DAVID M. CLARK

1. PRELIMINARIES

In this section we shall first make a few remarks on notation and ter-

minology, and then we shall review some well-known facts for the purpose of

reference. Given is a similarity type determined by a set Rl of relation

symbols and a set Op of operation symbols. A structure Oh has universe

A = I 011, and for each n-ary relation symbol R € Rl has an n-ary relation

R , and for each n-ary operation symbol f € Op has an n-ary operation f .

OL is called an algebra if Rl = 0 and 01 is called a relational struc-

ture if Op = 0.

If Wi is a class of structures then IVKI, SWt and H'Hx denote the

classes of isomorphic images, substructures and homomorphic images of members

of VYL respectively, and Finn, PVH, P inn and P yn denote the classes of

direct products, subdirect products, reduced direct products, and ultraproducts

of subsets of YV\ respectively. We allow the direct product of the empty set

of structures.

Next we introduce the (finitary) first-order language of the given

similarity type. Then AW?, 0»l and EWX denote the atomical, universal

and elementary classes generated by W\ respectively. For a class VU% of

algebras, AVH is called the variety generated by Wi and is denoted by VIM.

For any structure Ol , D is the trivial congruence on #L and ±

is the universal congruence on Ol .

Let *r c n0l. | i € I. For each F s I, TT_ is the projection of

1 r

n(Jl. | i € I onto T70l. I i € ¥, and 9 and 9** are the projection

congruences on T\01. | i € I and 5& - induced by TT_, respectively. 06- is

1 r

called non-trivially subdirect if c& is subdirect and for no i € I is

7T.: ^.- Ol . an isomorphism. If x, y € B then 9 (x, y) denotes the

smallest congruence x o n & such that xxy. A congruence x o n "6- is

called principal if there exist x, y € B such that x = 0 (x y)» If a is

an ordinal and x € (T7A. I i €

I)a

then TT_(X) € (ffA. j i € F )

a

is de-

1 r 1

fined by

TTF(X)(€) = TTF(X(C):

If cp is a formula and x € A^ then Ol \= cp[x] means that x satisfies

tp in OL , and we adopt the usual conventions to make satisfaction of for-

mulas by finite sequences well-behaved. If x € (flA. ) n then we define