- (建造物の)基礎,土台

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/08/28 16:46:57」(JST)

This article is about mathematics. For substructures of buildings in civil engineering, see Superstructure#Engineering .

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.

- 1 Definition
- 2 Example
- 3 Substructures as subobjects
- 4 Submodel
- 5 See also
- 6 References

Given two structures *A* and *B* of the same signature σ, *A* is said to be a **weak substructure** of *B*, or a **weak subalgebra** of *B*, if

- the domain of
*A*is a subset of the domain of*B*, *f*=^{A}*f*|^{B}*A*for every^{n}*n*-ary function symbol*f*in σ, and*R*^{A}*R*^{B}*A*for every^{n}*n*-ary relation symbol*R*in σ.

*A* is said to be a **substructure** of *B*, or a **subalgebra** of *B*, if *A* is a weak subalgebra of *B* and, moreover,

*R*=^{A}*R*^{B}*A*for every^{n}*n*-ary relation symbol*R*in σ.

If *A* is a substructure of *B*, then *B* is called a **superstructure** of *A* or, especially if *A* is an induced substructure, an **extension** of *A*.

In the language consisting of the binary functions + and ×, binary relation <, and constants 0 and 1, the structure (**Q**, +, ×, <, 0, 1) is a substructure of (**R**, +, ×, <, 0, 1). More generally, the substructures of an ordered field (or just a field) are precisely its subfields. Similarly, in the language (×, ^{-1}, 1) of groups, the substructures of a group are its subgroups. In the language (×, 1) of monoids, however, the substructures of a group are its submonoids. They need need not be groups; and even if they are groups, they need not be subgroups.

In the case of graphs (in the signature consisting of one binary relation), subgraphs, and its weak substructures are precisely its subgraphs.

For every signature σ, induced substructures of σ-structures are the subobjects in the concrete category of σ-structures and strong homomorphisms (and also in the concrete category of σ-structures and σ-embeddings). Weak substructures of σ-structures are the subobjects in the concrete category of σ-structures and homomorphisms in the ordinary sense.

In model theory, given a structure *M* which is a model of a theory *T*, a **submodel** of *M* in a narrower sense is a substructure of *M* which is also a model of *T*. For example if *T* is the theory of abelian groups in the signature (+, 0), then the submodels of the group of integers (**Z**, +, 0) are the substructures which are also groups. Thus the natural numbers (**N**, +, 0) form a substructure of (**Z**, +, 0) which is not a submodel, while the even numbers (2**Z**, +, 0) form a submodel which is (a group but) not a subgroup.

Other examples:

- The algebraic numbers form a submodel of the complex numbers in the theory of algebraically closed fields.
- The rational numbers form a submodel of the real numbers in the theory of fields.
- Every elementary substructure of a model of a theory
*T*also satisfies*T*; hence it is a submodel.

In the category of models of a theory and embeddings between them, the submodels of a model are its subobjects.

- Elementary substructure
- End extension
- Löwenheim-Skolem theorem
- Prime model

- Burris, Stanley N.; Sankappanavar, H. P. (1981),
*A Course in Universal Algebra*, Berlin, New York: Springer-Verlag - Diestel, Reinhard (2005) [1997],
*Graph Theory*, Graduate Texts in Mathematics**173**(3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-26183-4 - Hodges, Wilfrid (1997),
*A shorter model theory*, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

- 1. フィブロネクチン糸球体症 fibronectin glomerulopathy
- 2. 歯の解剖学的構造および発達 anatomy and development of the teeth
- 3. 混合型クリオグロブリン血症症候群 (本態性混合型クリオグロブリン血症)の臨床症状

および診断 clinical manifestations and diagnosis of the mixed cryoglobulinemia syndrome essential mixed cryoglobulinemia - 4. 米国における労働衛生および環境衛生に関する問題についての情報および教材 information and educational resources for occupational and environmental health issues in the united states

- Fabrication of porous chitin with continuous substructure by regeneration from gel with CaBr2·2H2O/methanol.

- Kadokawa J1, Endo R2, Tanaka K2, Ohta K2, Yamamoto K2.
- International journal of biological macromolecules.Int J Biol Macromol.2015 Jul;78:313-7. doi: 10.1016/j.ijbiomac.2015.04.022. Epub 2015 Apr 21.
- In this study, we investigated the fabrication of porous chitins with continuous channel substructure by regeneration from gels with CaBr2·2H2O/methanol solution. After rapidly removing methanol from the gels, the products were immersed in methanol, followed by washing out CaBr2 with water and lyop
- PMID 25910644

- Photodegradation mechanism of sulfonamides with excited triplet state dissolved organic matter: a case of sulfadiazine with 4-carboxybenzophenone as a proxy.

- Li Y1, Wei X1, Chen J2, Xie H1, Zhang YN1.
- Journal of hazardous materials.J Hazard Mater.2015 Jun 15;290:9-15. doi: 10.1016/j.jhazmat.2015.02.040. Epub 2015 Feb 16.
- Excited triplet states of dissolved organic matter ((3)DOM*) are important players for photodegradation sulfonamide antibiotics (SAs) in sunlit natural waters. However, the triplet-mediated reaction mechanism was poorly understood. In this study, we investigated the reaction adopting sulfadiazine as
- PMID 25731147

- A novel, soluble compound, C25, sensitizes to TRAIL-induced apoptosis through upregulation of DR5 expression.

- James MA1, Seibel WL, Kupert E, Hu XX, Potharla VY, Anderson MW.
- Anti-cancer drugs.Anticancer Drugs.2015 Jun;26(5):518-30. doi: 10.1097/CAD.0000000000000213.
- The tumor necrosis factor-related apoptosis-inducing ligand (TRAIL) is a potential therapeutic agent that induces apoptosis selectively in tumor cells. However, numerous solid tumor types are resistant to TRAIL. Sensitization to TRAIL has been an area of great research interest, but has met signific
- PMID 25646742

- Deformation Microstructure and Fracture Behavior in Creep-Exposed Alloy 617

- Materials transactions 58(3), 442-449, 2017-03
- NAID 40021118699

- 鉄骨系下部構造を有する円筒ラチスシェル体育館の耐震性能評価

- 日本建築学会東海支部研究報告集 Proceedings of Tokai Chapter Architectural Research Meeting (55), 109-112, 2017-02
- NAID 40021083152

- Search for anomalous electroweak production of WW/WZ in association with a high-mass dijet system in pp collisions at √s=8 TeV with the ATLAS detector

- Physical review D 95(3), 032001, 2017-02
- NAID 120006023406

- 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 ...

- In computer science, a problem is said to have optimal substructure if an optimal solution can be constructed efficiently from optimal solutions of its subproblems. This property is used to determine the usefulness of dynamic programming and ...

リンク元 | 「substructural」「下部構造」 |

- 下部構造の