Abstract. Given a class of first order structures ${\cal K}$ and an algebra $A$ of the type of ${\cal K}$, we define the set of $A${\it structures in ${\cal K}$}, in symbols ${\cal K}_A$, as the set of members of ${\cal K}$ whose underlying algebra is $A$. In this paper we pose the following problem: When the closure of ${\cal K}$ under the common operations (taking substructures, direct products,$\ldots $) can be expressed in terms of local properties concerning the sets ${\cal K}_A$ and global properties relating ${\cal K}_A$ and ${\cal K}_B$ whenever there is an algebra homomorphism from $A$ into $B$. We obtain some results in this direction which hold, in particular, for classes of structures axiomatized by equalityfree sentences. The fundamental result says that the classes ${\cal K}$ for which ${\cal K}_A$ is an algebraic closure system, for all algebras $A$, are precisely the (equalityfree) strict universal Horn classes.
AMS Subject Classification
(1991): 03C52, 03C30
Keyword(s):
Set of structures on an algebra,
filter extension,
local and global properties,
closure condition
Received December 23, 1997 and in final form August 31, 1998. (Registered under 2669/2009.)
