Mathematical logic | Model theory | Universal algebra

Substructure (mathematics)

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 restricted to the substructure's domain. 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. (Wikipedia).

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From playlist Real Analysis

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From playlist Addition and Subtraction of Whole Numbers

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From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)

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From playlist Prealgebra Chapter 1 (Complete chapter)

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From playlist Foundational Math

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From playlist Mathematics (All Of It)

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From playlist MIT Linear Finite Element Analysis

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From playlist Mathematics

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From playlist Prealgebra Chapter 1 (Complete chapter)

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From playlist Injective, Surjective, and Bijective Functions

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From playlist Behind the Scenes in Real-Life Software Design

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From playlist Logic and Foundations

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The Book of Proof: https://www.people.vcu.edu/~rhammack/BookOfProof ⭐Support the channel⭐ Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math My amazon shop: https://www.amazon.com/shop/michaelpenn 🟢 Discord: https://discord.gg/Ta6PTGtKB

From playlist Proof Writing

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From playlist Equations and Inequalities

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Haggai Maron: Subgraph-based networks for expressive, efficient, and domain-independent graph...

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From playlist Mathematical Aspects of Computer Science

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The Magical Gluing Lemma, Part 1

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From playlist The Magical Gluing Lemma

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From playlist Talks of Mathematics Münster's reseachers

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From playlist Spatial Data Analytics and Modeling

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From playlist Mathematics

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Graph (discrete mathematics) | Algebraically closed field | Subring | Glossary of graph theory | Group (mathematics) | Prime model | Löwenheim–Skolem theorem | Subobject | Model theory | Algebra over a field | Field extension | Field (mathematics) | Embedding | Category (mathematics) | Subset | Concrete category | Signature (logic) | Structure (mathematical logic) | Mathematical logic | Subgroup | End extension | Ordered field