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).

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).

Title subalgebra of an algebraic system
Canonical name SubalgebraOfAnAlgebraicSystem
Date of creation 2013-03-22 16:44:19
Last modified on 2013-03-22 16:44:19
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 08A30
Classification msc 08A05
Classification msc 08A62
Synonym subalgebra lattice
Defines subalgebra
Defines generating set
Defines subalgebra generated by
Defines extension of an algebraic system
Defines restriction
Defines proper subalgebra
Defines lattice of subalgebras
Defines spanning set
Defines finitely generated
Defines cyclic