Algebraic Structure

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An Algebraic Structure is a mathematical system composed of a set structure together with some set of finitary operations.



  • (Wikipedia, 2016) ⇒ Retrieved:2016-3-31.
    • In mathematics, and more specifically in abstract algebra, the term algebraic structure generally refers to a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. [1]

      Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras.

      The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. Category theory is used to study the relationships between two or more classes of algebraic structures, often of different kinds. For example, Galois theory studies the connection between certain fields and groups, algebraic structures of two different kinds.

  1. P.M. Cohn. (1981) Universal Algebra, Springer, p. 41.


    • An algebraic system, loosely speaking, is a set, together with some operations on the set. Before formally defining what an algebraic system is, let us recall that a $n$-ary operation (or operator) on a set $A$ is a function whose domain is $A^n$ and whose range is a subset of $A$. Here, $n$ is a non-negative integer. When $n=0$, the operation is usually called a nullary operation, or a constant, since one element of $A$ is singled out to be the (sole) value of this operation. A finitary operation on $A$ is just an $n$-ary operation for some non-negative integer $n$.

      Definition. An algebraic system is an ordered pair $(A,O)$, where $A$ is a set, called the underlying set of the algebraic system, and $O$ is a set, called the operator set, of finitary operations on $A$.

      We usually write $\boldsymbol{A}$, instead of $(A,O)$, for brevity.

      A prototypical example of an algebraic system is a group, which consists of the underlying set $G$, and a set $O$ consisting of three operators: a constant $e$ called the multiplicative identity, a unary operator called the multiplicative inverse, and a binary operator called the multiplication.