Stable group

This article defines a property of a group (possibly with additional operations and structure) as viewed in logic/model theory

Definition

Definition with symbols

A group (possibly with additional structure and relations) is said to be stable if there is no formula $f({\overline {x}},{\overline {y}})$ in the theory of the group that can order an infinite set of tuples. By compactness, this is equivalent to saying that there is no formula that cna order arbitrarily large finite sets of tuples. This characterization of stability is termed the order property.

Note that putting in more additional structure can only increase the number of formulae possible, and hence a group which is stable under some additional structures and relations, is also stable as a pure group.