ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

43. Theorem: Substructure is a structure of same type
Let B be a substructure of A. When B is equipped with the induced operations
you get a structure of same type as A. We shall always let B also denote this
structure if no other structure is explicitly specified.
Substructures occur in a lot of situations.
44. Theorem: The generated substructure
Let (A, ♢ 1
, . . . , ♢ n
) be a algebraic structure and let D ⊂ A. Then there exists
a substructure B that contains D and is the least one to do so, which means
that it is contained in any other substructure containing D.
It is the intersection of all substructures containing D and can be explicitly
written as
B = ∩ {C ∈ A : D ⊆ C}
where A denotes the set of substructures of (A, ♢ 1
, . . . , ♢ n
Proof : We define B by the formula and starts by showing that B is in fact a
substructure. (Remark that A ∈ A and so A is not empty). To do so we must
show that B is invariant wrt any operation.
So let ♢ be any operation of the structure, say with arity k. Let (b1, . . . , bk) ∈
B k . We must show that ♢ (b1, . . . , bk) ∈ B. So let C be any substructure
containing D. Since C is a substructure we have that ♢ (b1, . . . , bk) ∈ C.
Since this is true for any C we have that ♢ (b1, . . . , bk) must also be in the
intersection of all these C which is B.
By construction B is contained in any substructure containing C
45. Definition: The generated substructure, the algebraic closure
The structure B in the preceding theorem is said to be the subalgebra generated
by D, or the algebraic closure of D. It is denoted by ⟨D⟩. If D = {d1, . . . , dn}
we shall also write ⟨D⟩ = ⟨d1, . . . , dn⟩
46. Theorem: A characterization of the algebraic closure
The algebraic closure ⟨D⟩ consists of all elements which can be obtained by
successive application of the operations on elements of D
28
).