Homogeneous Structure

A homogeneous structure is a model in which every isomorphism between finite (or otherwise small, by context) substructures extends to an automorphism of the whole structure; equivalently, tuples with the same quantifier-free (or complete) type over the empty set lie in the same orbit under the automorphism group, exhibiting high internal symmetry.

A structure is called homogeneous (often ultrahomogeneous) if every isomorphism between its finite substructures extends to an automorphism of the whole structure. Equivalently, finite patterns that occur in the structure can be moved anywhere by global symmetries.