In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure.

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  • In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure. In model theory, the term "submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models. In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are at most those induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures. Ordered groups, on the other hand, have the special property that every substructure of an ordered group which is itself an ordered group, is an induced substructure.
  • 在数学学科模型论中，某个其他模型的子模型或子结构是满足与最初模型同样关系的更小的模型。 形式定义如下。设 <math>M</math> 和 <math>N</math> 是同一个语言 <math>L</math> 的两个模型。我们称 <math>M</math> 是 <math>N</math> 的子模型(通常表示为 M ⊂ N) (等价的说，<math>N</math> 是 <math>M</math>的扩展)当且仅当 <math>M</math> 的域是 <math>N</math> 的域的子集； 对于所有 <math>L</math> 的 <math>n</math>-元关系符号 <math>R</math>，我们有 R = R ∩ M； 对于所有 <math>L</math> 的 <math>m</math>-元函数符号 <math>f</math>，我们有 <math>f^M = f^N|M^m \ </math>； 对于所有 <math>L</math> 的常量符号 <math>c</math>，我们有 <math>c^M = c^N \ </math>。 比如 (Q, +, ×, <, 0, 1) 是 (R, +, ×, <, 0, 1) 的子模型。 在语言的模型的范畴中，子模型将是子对象。
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  • In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure.
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  • Substructure
  • 子结构
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