Algebraic structure

https://en.wikipedia.org/wiki/Algebraic_structure

Algebraic structure

An algebraic structure is a nonempty set M together with one or more operations (i.e. a function ∗:M×M→M) which satisfy some axioms. In other words, the definition of an algebraic structure is an axiomatic system. It can be proved that an axiomatic system which defines an algebraic structure verifies the logic requirements which any axiomatic system has to fulfill.

All I can do is give you some pointers to the standard terminology of elementary model theory and universal algebra: a structure consists of a set equipped with a set of finitary operations and relations (https://en.wikipedia.org/wiki/Structure_(mathematical_logic). An algebraic structure is one in which the only relation is equality (which many authors treat as a logical rather than mathematical concept) (https://en.wikipedia.org/wiki/Algebraic_structure). It is usual to associate a fixed set names for the operations and relations used in a structure. Such an association is called a signature for the structure and results in a logical language with function and relation symbols for the operations and relations. An axiomatic system is just a set of axioms in a logical language (https://en.wikipedia.org/wiki/Axiomatic_system). Any set of axioms in the language defined by some signature defines a class of structures for its signature (namely the structures in which those axioms hold) and conversely a class of structures for a signature defines a set of axioms (namely the axioms which hold in each structure in the class). A class of algebraic structures defined by a set of equational axioms is called an algebraic variety (https://en.wikipedia.org/wiki/Algebraic_variety).

A poset has a structure, but not an algebraic structure. A poset is equipped with a partial order relation, this is not a finitary operation. For a set S an n-tary operation is a function F:S^n → S. An order relation is not a function.

An algebraic structure is a totally ordered set, whose elements are

• sets S_i,

• (finitary) operations O_j over these sets,

• relations R_k between these sets.

An example of structure with more than one set is a vector space W over a field K: (K,W,+,×). An example of structure with also a relation is an ordered field: (K,+,⋅,≤)

Algebraic structure is a carrier set together with operations on that carrier set, with the operations satisfying certain axioms.

A set with such an additional structure is also called an algebra. Some algebraic structures have friendly names, such as group, field, ring.

An algebraic structure, or algebra, is a type of mathematical object, more narrowly, it is a special mathematical structure. To satisfy the minimal requirements of becoming an algebraic structure, a mathematical structure must be equipped with a binary operation and obey a certain set of axioms.

3 parts of an algebraic structure: 1. a carrier set 2. a set of operations 3. a set of axioms

The underlying set

A carrier (or underlying) set is a plain old set. For example, the set of natural numbers. In programming languages, a type is the underlying set. This set alone is never an algebra unless it has an additional structure "attached".

The operations

Only when a carrier set is equipped with an operation and a set of axioms, it becomes an algebra. The operation that operates on (combines, relates) the elements of the set must satisfy the prescribed axioms.

Commonly, the operation is binary and denoted by a ⋆ ("star") or • ("dot").

An example

A familiar group is the set of integers, $\mathbb{Z}$, together with addition (denoted by a plus sign), $G = \{ \mathbb{Z}, +\}$. This group upholds the axioms of totality, identity, associativity, invertibility and commutativity. Since commutativity is not the required group axiom, this is an abelian or commutative group.

The integers are closed under addition. There is a unique identity element, 0. Every integer has an inverse; when an integer is combined with its inverse, the identity element is produced; e.g. inverse pairs: (1,-1), (-1,1), (0,0).

$\forall x,y,z \in \mathbb{Z}$:

• closure: $x+y \in \mathbb{Z}$

• identity: $x+0 = 0+x = x$

• invertibility: $x+(-x) = (-x) + x = 0$

• associativity: $x+(y+z) = (x+y)+z$

• commutativity: $x+y = y+x$