
In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. 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 and structurepreserving map . The precise meaning of "structurepreserving" 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 theory due to D. B. McAlister states that every inverse semigroup has an Eunitary cover; besides being surjective, the homomorphism in this case is also ''idempotent separating'', meaning that in its kernel an idempotent and nonidempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an Finverse cover.〔Lawson p. 230〕 McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover.〔Grilett p. 360〕 Examples from other areas of algebra include the Frattini cover of a profinite group and the universal cover of a Lie group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cover (algebra)」の詳細全文を読む スポンサード リンク
