subalgebra of an algebraic system
Let be an algebraic system ( is the underlying set and is the set of operators on ).
Subalgebras of an Algebra
Let be a non-empty subset of . is closed under operators of if for each -ary operator on , and any , we have .
Suppose is closed under operators of . For each -ary operator on , we define by . Each of these operators is well-defined and is called a restriction (of the corresponding ). Furthermore, is a well-defined algebraic system, and is called the subalgebra of . When is a subalgebra of , we also say that is an extension of .
is clearly a subalgebra of itself. Any other subalgebra of is called a proper subalgebra.
Remark. If contains constants, then any subalgebra of must contain the exact same constants. For example, the ring of integers is an algebraic system with no proper subalgebras. Indeed, if is a subring of , , so .
Since we are operating under the same operator set, we can, for convenience, drop and simply call an algebra, a subalgebra of , etc… If are subalgebras of , then is also a subalgebra. In fact, given any set of subalgebras of , their intersection is also a subalgebra.
Generating Set of an Algebra
Let be any subset of an algebra . Consider the collection of all subalgebras of containing . This collection is non-empty because . The intersection of all these subalgebras is again a subalgebra containing the set . Denote this subalgebra by . is called the subalgebra spanned by , and is called the spanning set of . Conversely, any subalgebra of has a spanning set, namely itself: .
Given a subalgebra of , a minimal spanning set of is called a generating set of . By minimal we mean that the set obtained by deleting any element from no longer spans . When has a generating set , we also say that generates . If can be generated by a finite set, we say that is finitely generated. If can be generated by a single element, we say that is cyclic.
Remark. the subalgebra generated by the constants of . If no such constants exist, .
From the discussion above, the set of subalgebras of an algebraic system forms a complete lattice. Given subalgebras , is the intersection of all , and is the subalgebra . The lattice of all subalgebras of is called the subalgebra latttice of , and is denoted by .
|Title||subalgebra of an algebraic system|
|Date of creation||2013-03-22 16:44:19|
|Last modified on||2013-03-22 16:44:19|
|Last modified by||CWoo (3771)|
|Defines||subalgebra generated by|
|Defines||extension of an algebraic system|
|Defines||lattice of subalgebras|