Saying "one-to-one and onto correspondence" is not necessary because a one-to-one correspondence is defined as a one-to-one and onto.

Define to be the natural projection. If is a subring of then is a subring of . If is a subring of then is a subring of which contains .

Let be the set of all subrings of which contain and be the set of all subrings of . By above paragraph define as for all .