Algebraic Structures
An operation (more precisely, a binary operation) on a set S is a function from S × S to S. Standard examples are addition, multiplication, and composition of functions. Elementary texts often emphasize the "closure" property of an operation (or, sometimes, of an algebraic structure): the...
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| Vydáno v: | A Guide to Groups, Rings, and Fields s. 17 - 28 |
|---|---|
| Hlavní autor: | |
| Médium: | Kapitola |
| Jazyk: | angličtina |
| Vydáno: |
Washington
The Mathematical Association of America
2012
Mathematical Association of America American Mathematical Society |
| Vydání: | 1 |
| Témata: | |
| ISBN: | 9780883853559, 0883853558 |
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| Abstract | An operation (more precisely, a binary operation) on a set S is a function from S × S to S. Standard examples are addition, multiplication, and composition of functions.
Elementary texts often emphasize the "closure" property of an operation (or, sometimes, of an algebraic structure): the product of two elements in S must be an element of S. We have, instead, built this into the definition.
An algebraic structure (Bourbaki says a magma) is a set equipped with one or more operations. Such structures sometimes come with distinguished elements (such as identity elements) or functions associated with the operation (such as taking inverses).
An algebraic structure is said to be finite if the underlying set S is finite. We will write |S| for the number of elements of S, which is often referred to as the order of S.
For each kind of algebraic structure there is a corresponding choice of "good functions" from one object to another, usually those that preserve the structure. These are usually called homomorphisms.
Attempts have been made to produce a general theory of algebraic structures, for example in "universal algebra." Some good references are [43, ch. 2] and [29].
STRUCTURES WITH ONE OPERATION
Suppose we have a set S with one operation,which we will denote by juxtaposition, (a, b) ↦ ab, and call the "product of a and b" unless there is risk of confusion. |
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| AbstractList | An operation (more precisely, a binary operation) on a set S is a function from S × S to S. Standard examples are addition, multiplication, and composition of functions.
Elementary texts often emphasize the "closure" property of an operation (or, sometimes, of an algebraic structure): the product of two elements in S must be an element of S. We have, instead, built this into the definition.
An algebraic structure (Bourbaki says a magma) is a set equipped with one or more operations. Such structures sometimes come with distinguished elements (such as identity elements) or functions associated with the operation (such as taking inverses).
An algebraic structure is said to be finite if the underlying set S is finite. We will write |S| for the number of elements of S, which is often referred to as the order of S.
For each kind of algebraic structure there is a corresponding choice of "good functions" from one object to another, usually those that preserve the structure. These are usually called homomorphisms.
Attempts have been made to produce a general theory of algebraic structures, for example in "universal algebra." Some good references are [43, ch. 2] and [29].
STRUCTURES WITH ONE OPERATION
Suppose we have a set S with one operation,which we will denote by juxtaposition, (a, b) ↦ ab, and call the "product of a and b" unless there is risk of confusion. Anoperation(more precisely, abinary operation) on a setSis a function fromS × StoS. Standard examples are addition, multiplication, and composition of functions. Elementary texts often emphasize the “closure” property of an operation (or, sometimes, of an algebraic structure): the product of two elements inSmust be an element ofS. We have, instead, built this into the definition. Analgebraic structure(Bourbaki says amagma) is a set equipped with one or more operations. Such structures sometimes come with distinguished elements (such as identity elements) or functions associated with the operation (such |
| Author | Fernando Q. Gouvêa |
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| Keywords | oid-construction ring abelian group magma idempotent rig associative algebra partially ordered set algebraic structure dioid limit action associative join closure universal algebra group equivariant module algebraic structure finite rng tropical semiring unit field linear transformation totally ordered set homomorphism skew field vector space bad habits groupoid algebra representation ordered set identity element supremum octonions poset Jacobi identity monoidoid nonassociative "ring" inverse hypercomplex numbers semiring division ring nonassociative operations action via monoid ring homomorphism infimum meet preordered set zero ring associative ring lattice semigroup Lie algebra operation commutative |
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| Snippet | An operation (more precisely, a binary operation) on a set S is a function from S × S to S. Standard examples are addition, multiplication, and composition of... Anoperation(more precisely, abinary operation) on a setSis a function fromS × StoS. Standard examples are addition, multiplication, and composition of... |
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| SubjectTerms | Abstract algebra Abstract spaces Algebra Algebraic structure Applied mathematics Applied statistics Category theory Dimensional analysis Dimensionality Formal logic Homomorphisms Logic Logical topics Mathematical logic Mathematical objects Mathematical rings Mathematical set theory Mathematical sets Mathematics Monoids Morphisms Partially ordered sets Philosophy Pure mathematics Semigroups Statistical physics Statistics Vector spaces |
| Title | Algebraic Structures |
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