ACTA issues

## Local characterization theorems for some classes of structures

Janusz Czelakowski, Raimon Elgueta

Acta Sci. Math. (Szeged) 65:1-2(1999), 19-32
2669/2009

 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 equality-free 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 (equality-free) 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.)