lattice of ideals
is a complete lattice.
First, is a left ideal of : if , then for all . Consequently, and so . Furthermore, if , then for any , so also. Hence is a left ideal. By construction, is clearly contained in all of , and is clearly the largest such ideal.
For the second part, we want to show that actually exists for arbitrary . We know the existence of if is finite. Suppose now is infinite. Define to be the set of finite sums of elements of . If , then , being a finite sum itself, clearly belongs to . Also, as well, since the additive inverse of each of the additive components of is an element of . Now, if , then too, since multiplying each additive component of by (on the left) lands back in . So is a left ideal. It is evident that . Also, if is a left ideal containing each , then any finite sum of elements of must also be in , hence . This implies that is the smallest ideal containing each of the . Therefore exists and is equal to .
From the above proof, we see that the sum of ideals can be equivalently interpreted as
the “ideal” of finite sums of the elements of , or
the “ideal” generated by (elements of) , or
the join of ideals .
Looking more closely at the above proof, we also have the following:
is an algebraic lattice.
However, the more general case is not true: the set of order ideals in a poset is a dcpo.
|Title||lattice of ideals|
|Date of creation||2013-03-22 16:59:40|
|Last modified on||2013-03-22 16:59:40|
|Last modified by||CWoo (3771)|