abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, a cover is one instance of some

mathematical structure In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...

mapping

onto In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

another instance, such as a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

(trivially) covering a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

. This should not be confused with the concept of a cover in topology. When some object ''X'' is said to cover another object ''Y'', the cover is given by some

surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ''X'' and ''Y'' are instances. In order to be interesting, the cover is usually endowed with additional properties, which are highly dependent on the context.

Examples

A classic result in

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...

theory due to

D. B. McAlister D. or d. may refer to, usually as an abbreviation: * Don (honorific), a form of address in Spain, Portugal, Italy, and their former overseas empires, usually given to nobles or other individuals of high social rank. * Date of death, as an abbreviati ...

states that every

inverse semigroup In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigr ...

has an E-unitary cover; besides being surjective, the homomorphism in this case is also ''

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

separating'', meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover. McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover. Examples from other areas of algebra include the

Frattini cover Frattini is an Italian surname. Notable people with the surname include: * Angelo Frattini, sculptor * Franco Frattini, politician * Giovanni Frattini, mathematician ** Frattini argument ** Frattini subgroup * Francesco Frattini, cyclist * P ...

of a

profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...

and the

universal cover In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...

of a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

Modules

If ''F'' is some family of modules over some ring ''R'', then an ''F''-cover of a module ''M'' is a homomorphism ''X''→''M'' with the following properties: *''X'' is in the family ''F'' *''X''→''M'' is surjective *Any surjective map from a module in the family ''F'' to ''M'' factors through ''X'' *Any endomorphism of ''X'' commuting with the map to ''M'' is an automorphism. In general an ''F''-cover of ''M'' need not exist, but if it does exist then it is unique up to (non-unique) isomorphism. Examples include: *

Projective cover In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes. Definition ...

s (always exist over

perfect ring In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring over which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, ther ...

s) *

flat cover In algebra, a flat cover of a module ''M'' over a ring is a surjective homomorphism from a flat module ''F'' to ''M'' that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers a ...

s (always exist) *

torsion-free cover In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule contains only t ...

s (always exist over integral domains) *

injective cover In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

Notes

References

* Abstract algebra {{abstract-algebra-stub

Examples

Modules

See also

Notes

References