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单词 SubalgebraOfAnAlgebraicSystem
释义

subalgebra of an algebraic system


Let (A,O) be an algebraic system (A is the underlying set and O is the set of operators on A).

SubalgebrasMathworldPlanetmathPlanetmathPlanetmath of an AlgebraMathworldPlanetmath

Let B be a non-empty subset of A. B is closed under operators of A if for each n-ary operator ωA on A, and any b1,,bnB, we have ωA(b1,,bn)B.

Suppose B is closed underPlanetmathPlanetmath operators of A. For each n-ary operator ωA on A, we define ωB:BnB by ωB(b1,,bn):=ωA(b1,,bn). Each of these operators is well-defined and is called a restrictionPlanetmathPlanetmath (of the corresponding ωA). Furthermore, (B,O) is a well-defined algebraic system, and is called the subalgebra of (A,O). When (B,O) is a subalgebra of (A,O), we also say that (A,O) is an extensionPlanetmathPlanetmathPlanetmath of (B,O).

(A,O) is clearly a subalgebra of itself. Any other subalgebra of (A,O) is called a proper subalgebra.

Remark. If (A,O) contains constants, then any subalgebra of (A,O) must contain the exact same constants. For example, the ring of integers is an algebraic system with no proper subalgebras. Indeed, if R is a subring of , 1R, so R=.

Since we are operating under the same operator set, we can, for convenience, drop O and simply call A an algebra, B a subalgebra of A, etc… If B1,B2 are subalgebras of A, then B1B2 is also a subalgebra. In fact, given any set of subalgebras Bi of A, their intersectionMathworldPlanetmath Bi is also a subalgebra.

Generating Set of an Algebra

Let C be any subset of an algebra A. Consider the collectionMathworldPlanetmath [C] of all subalgebras of A containing C. This collection is non-empty because A[C]. The intersection of all these subalgebras is again a subalgebra containing the set C. Denote this subalgebra by C. C is called the subalgebra spanned by C, and C is called the spanning set of C. Conversely, any subalgebra B of A has a spanning set, namely itself: B=B.

Given a subalgebra B of A, a minimalPlanetmathPlanetmath spanning set X of B is called a generating set of B. By minimal we mean that the set obtained by deleting any element from X no longer spans B. When B has a generating set X, we also say that X generates B. If B can be generated by a finite setMathworldPlanetmath, we say that B is finitely generatedMathworldPlanetmathPlanetmath. If B can be generated by a single element, we say that B is cyclic.

Remark. = the subalgebra generated by the constants of A. If no such constants exist, :=.

From the discussion above, the set of subalgebras of an algebraic system forms a complete latticeMathworldPlanetmath. Given subalgebras Ai, Ai is the intersection of all Ai, and Ai is the subalgebra Ai. The latticeMathworldPlanetmath of all subalgebras of A is called the subalgebra latttice of A, and is denoted by Sub(A).

Titlesubalgebra of an algebraic system
Canonical nameSubalgebraOfAnAlgebraicSystem
Date of creation2013-03-22 16:44:19
Last modified on2013-03-22 16:44:19
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id9
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 08A30
Classificationmsc 08A05
Classificationmsc 08A62
Synonymsubalgebra lattice
Definessubalgebra
Definesgenerating set
Definessubalgebra generated by
Definesextension of an algebraic system
Definesrestriction
Definesproper subalgebra
Defineslattice of subalgebras
Definesspanning set
Definesfinitely generated
Definescyclic

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更新时间:2025/5/4 4:54:44