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노트

위키데이터

• ID : Q205464

말뭉치

1. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure.
2. In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition.
3. Loosely speaking, an algebraic structure is any set upon which "arithmetic-like operations have been defined.
4. Most of the important sets in Linear Algebra possess some type of algebraic structure, and abelian groups are the principal building block of virtually every one of these algebraic structures.
5. This is because the field axioms are extremely specific in describing algebraic structure.
6. A group is always a monoid, semigroup, and algebraic structure.
7. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures.
8. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure.
9. In this way, every algebraic structure gives rise to a category.
10. Knowing the name ‘algebraic structure’ helps protect yourself from that.
11. Hence why they called the algebraic structure specification for JavaScript ‘Fantasy Land’.
12. A set with one or more binary operations gives rise to what is commonly known as an algebraic structure.
13. In particular, the set Z of integers under the addition ‘+’ is an algebraic structure.
14. In the same way, the set of rational numbers Q under the usual multiplication operation ‘x’, and denoted by (Q, x), is another algebraic structure.
15. Such an algebraic structure is denoted by (R, +, x).
16. An algebraic structure is a collection of objects and operations that can be used to calculate and solve equations.
17. A non-empty set G equipped with one or more binary operations is said to be an algebraic structure.
18. Suppose * is a binary operation on G. Then (G, *) is an algebraic structure.
19. By a property of an algebraic structure, we mean a property possessed by any of its operations.
20. There are several notions of an algebraic structure on an object of some category or higher category, which differ in generality.
21. We say that a functor in the base category preserves some algebraic structure if it lifts to the corresponding category of algebras.
22. In mathematics, and more specifically in abstract algebra, the term algebraic structure generally refers to an arbitrary set with one or more finitary operations defined on it.
23. For another example, the group can be seen as a set that is equipped with an algebraic structure, namely the operation .
24. We have a set, however to have algebraic structure we need one thing more.
25. For instance int set has operation of adding elements, then int set with adding operation - binary function int + int -> int , creates an algebraic structure, and in this example it is Semigroup.
26. Indeed, binary trees can represent expressions, and they form an algebraic structure.
27. The emphasis is on the algebraic nature of real automation, which appears as a natural three-sorted algebraic structure, that allows for a rich algebraic theory.

메타데이터

위키데이터

• ID : Q205464

Spacy 패턴 목록

• [{'LOWER': 'algebraic'}, {'LEMMA': 'structure'}]