structureconsecutio

Structureconsecutio is a term used in theoretical discussions to describe a sequence of mathematical structures of a fixed signature arranged in a directed order, together with structure-preserving maps between successive members. The term is not standard in published literature; it is derived from Latin consecutio, meaning succession, and is used informally to emphasize the consecutive extension of structure from one stage to the next.

Formally, a structureconsecutio (A_i)_{i∈I} consists of an index set I with a directed order, a family of L-structures A_i for a fixed language L, and embeddings e_i^{i+1}: A_i → A_{i+1} that preserve all symbols. Often one requires that A_i is a substructure of A_{i+1} (i.e., e_i^{i+1} is a substructure or elementary embedding). In the continuous case, the union A = ∪_{i∈I} A_i is called the direct limit or limit model, and each A_i embeds into A via the canonical maps.

Common examples include chains of finite graphs or groups with inclusion maps, or chains of models of a first-order theory under elementary embeddings. For instance, a simple graph chain may take P_n as a path on n vertices with P_n ⊆ P_{n+1}. In model theory, an elementary chain M_0 ≤ M_1 ≤ M_2 ≤ ... with union M is a model of T if each M_i is a model of T and each embedding is elementary.

Structureconsecutio is closely related to filtrations, inductive limits, and directed systems of structures. It emphasizes the constructive perspective of building complex objects by successive, structure-preserving extensions.