homogeneoussuch

Homogeneoussuch is a neologism used in some mathematical discussions to describe a uniformity property of a structure with respect to a specified language or set of relations. It is not a standard term in mainstream literature, where the conventional notions of homogeneity or ultrahomogeneity are typically employed. The term is sometimes introduced to emphasize the idea of uniformity of local configurations across the entire structure.

Definition and formalism

In a language L, a structure M is said to be homogeneoussuch if, for every finite substructure A of M and every pair of L-embeddings e1, e2: A → M that preserve the interpretations of the symbols in L, there exists an automorphism f of M such that f ∘ e1 = e2. Equivalently, any two finite substructures of M that are isomorphic within M can be mapped onto each other by a global symmetry of M. This aligns closely with the standard notion of homogeneity used in model theory, though homogeneoussuch is sometimes used to draw attention to the “such that” or uniformity aspect of the extensions.

Relation to other concepts

The property described here overlaps with ultrahomogeneity (when the condition extends to all finite substructures for all isomorphisms) and with more general homogeneous structures in model theory. Classic examples include the Rado (random) graph and certain Fraïssé limits, which are ultrahomogeneous. In nonstandard contexts, a weaker form called plain homogeneity may be discussed, depending on which finite patterns are required to extend via automorphisms.

History and usage

Homogeneoussuch does not have wide formal adoption and is primarily encountered as a playful or clarifying synonym in informal expositions. For rigorous treatment, readers are directed to the established concepts of homogeneous and ultrahomogeneous structures in model theory. See also: homogeneous structure, ultrahomogeneous structure, Fraïssé limit.