Algebraic structure Totally Explained

Everything about Algebraic Structure totally explained

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra. Abstractly, an "algebraic structure" is the collection of all possible models of a given set of axioms. More concretely, an algebraic structure is any particular model of some set of axioms. For example, the monster group both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other groups. This article employs both meanings of "structure."
This definition of an algebraic structure shouldn't be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all groups are also semigroups and magmas.

Structures whose axioms are all identities

If the axioms defining a structure are all identities, the structure is a variety (not to be confused with algebraic variety in the sense of algebraic geometry). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra.
All structures in this section are varieties. Some of these structures are most naturally axiomatized using one or more nonidentities, but are nevertheless varieties because there exists an equivalent axiomatization, one perhaps less perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and differ from varieties in their metamathematical properties.
In this section and the following one, structures are listed in approximate order of increasing complexity, operationalized as follows:

Simple structures requiring but one set, the universe S, are listed before composite ones requiring two sets;
Structures having the same number of required sets are then ordered by the number of binary operations (0 to 4) they require. Incidentally, no structure mentioned in this entry requires an operation whose arity exceeds 2;
Let A and B be the two sets that make up a composite structure. Then a composite structure may include 1 or 2 functions of the form AxA→B or AxB→A;
Structures having the same number and kinds of binary operations and functions are more or less ordered by the number of required unary and 0-ary (distinguished elements) operations, 0 to 2 in both cases.

The indentation structure employed in this section and the one following is intended to convey information. If structure B is under structure A and more indented, then all theorems of A are theorems of B; the converse doesn't hold.
Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they're linked by the absorption law. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models. Simple structures: No binary operation:

Set: a degenerate algebraic structure having no operations.

Pointed set: S has one or more distinguished elements, often 0, 1, or both.

Unary system: S and a single unary operation over S.

Pointed unary system: a unary system with S a pointed set. Group-like structures:
One binary operation, denoted by concatenation. For monoids, boundary algebras, and sloops, S is a pointed set.

Magma or groupoid: S and a single binary operation over S.

Steiner magma: A commutative magma satisfying x(xy) = y.
- Squag: an idempotent Steiner magma.
- Sloop: a Steiner magma with distinguished element 1, such that xx = 1.
Semigroup: an associative magma.
- Monoid: a unital semigroup.
  - Group: a monoid with a unary operation, inverse, giving rise to an inverse element.
    - Abelian group: a commutative group.
  - Band: a semigroup of idempotents.
    - Semilattice: a commutative band. The binary operation can be called either meet or join.
      - Boundary algebra: a unital semilattice (equivalently, an idempotent commutative monoid) with a unary operation, complementation, denoted by enclosing its argument in parentheses, giving rise to an inverse element that's the complement of the identity element. The identity and inverse elements bound S. Also, x(xy) = x(y) holds.
Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they're (nonassociative) magmas. Quasigroups feature 3 binary operations only because establishing the quasigroup cancellation property by means of identities alone requires two binary operations in addition to the group operation.
Quasigroup: a cancellative magma. Equivalently, ∀x,y∈S, ∃!a,b∈S, such that xa = y and bx = y.
- Loop: a unital quasigroup with a unary operation, inverse.
  - Moufang loop: a loop in which a weakened form of associativity, (zx)(yz) = z(xy)z, holds.
    - Group: an associative loop.

Lattice: Two or more binary operations, including meet and join, connected by the absorption law. S is both a meet and join semilattice, and is a pointed set if and only if S is bounded. Lattices often have no unary operations. Every true statement has a dual, obtained by replacing every instance of meet with join, and vice versa.

Bounded lattice: S has two distinguished elements, the greatest lower bound and the least upper bound. Dualizing requires replacing every instance of one bound by the other, and vice versa.

Complemented lattice: a lattice with a unary operation, complementation, denoted by postfix ', giving rise to an inverse element. That element and its complement bound the lattice.

Modular lattice: a lattice in which the modular identity holds.

Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse doesn't hold.
- Kleene algebra: a bounded distributive lattice with a unary operation whose identities are x"=x, (x+y)'=x'y', and (x+x')yy'=yy'. See "ring-like structures" for another structure having the same name.
- Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
  - Interior algebra: a Boolean algebra with an added unary operation, the interior operator, denoted by postfix ' and obeying the identities x'x=x, x"=x, (xy)'=x'y', and 1'=1.
    - Relation algebra: an interior algebra whose interior operator is called converse. S is always the Cartesian square of some set, and is a monoid under an added residuated binary operation, relative product, whose identity element is distinct from the Boolean bounds. Relative product distributes over meet or join.
  - Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x'x=1, x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z).

Ringoids: Two binary operations, addition and multiplication, with multiplication distributing over addition. Semirings are pointed sets.

Semiring: a ringoid such that S is a monoid under each operation. Each operation has a distinct identity element. Addition also commutes, and has an identity element that annihilates multiplication.

Commutative semiring: a semiring with commutative multiplication.
Ring: a semiring with a unary operation, additive inverse, giving rise to an inverse element equal to the additive identity element. Hence S is an Abelian group under addition.
- Rng: a ring lacking a multiplicative identity.
- Commutative ring: a ring with commutative multiplication.
  - Boolean ring: a commutative ring with idempotent multiplication, equivalent to a Boolean algebra.
- Kleene algebra: a semiring with idempotent addition and a unary operation, the Kleene star, denoted by postfix * and obeying the identities (1+x*x)x*=x* and (1+xx*)x*=x*. See "Lattice-like structures" for another structure having the same name. N.B. The above definition of ring doesn't command universal assent. Some authorities employ "ring" to denote what is here called a rng, and refer to a ring in the above sense as a "ring with identity."

Modules: Composite Systems Defined over Two Sets, M and R:
The members of:

R are scalars, denoted by Greek letters. R is a ring under the binary operations of scalar addition and multiplication;

M are module elements (often but not necessarily vectors), denoted by Latin letters. M is an abelian group under addition. There may be other binary operations. The scalar product of scalars and module elements is a function RxM→M which commutes, associates (∀r,s∈R, ∀x∈M, r(sx) = (rs)x ), has 1 as identity element, and distributes over module and scalar addition. If only the pre(post)multiplication of module elements by scalars is defined, the result is a left (right) module.

Free module: a module having a free basis, where the operations are defined by performing the operation in Z, dividing by p and taking the remainder; see modular arithmetic.

Allowing additional structure

Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a topology. The added structure must be compatible, in some sense, with the algebraic structure.

Ordered group: a group with a compatible partial order. I.e., S is partially ordered.

Linearly ordered group: a group whose S is a linear order.

Archimedean group: a linearly ordered group for which the Archimedean property holds.

Lie group: a group whose S has a compatible smooth manifold structure.

Topological group: a group whose S has a compatible topology.

Topological vector space: a vector space whose M has a compatible topology; a superset of normed vector spaces.

Category theory

The discussion above has been cast in terms of elementary abstract and universal algebra. Category theory is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure.
There are various concepts in category theory that try to capture the algebraic character of a context, for instance

algebraic

essentially algebraic

presentable

locally presentable

monadic functors and categories

universal property.Further Information

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