Can transfinite numbers be extended to more general algebraic structures?
Infinite ordinals and cardinals are beautiful mathematical objects, extending in a natural way and sharing most of its properties. I have always wondered whether it would be possible to generalize the constructions of or to get transfinite integers or rationals. In general, putting a group structure on classes extending or is not possible as I indicated some years ago. Basically, we would have (and also ) which would imply . However, we can consider a slightly modified problem with weaker constraints, as follows. First we assume we have a class of cardinals containing zero and stable by addition.
We define the class , where is a representation of "negative" cardinals. As we have seen, the class need not be a group. Nevertheless, we can try to find an equivalence class over such that is a group. Moreover, we want this equivalence class to be compatible with addition and opposite i.e.
Of course this implies that is the identity element of the expected group. Similarly, we can settle the same problem for a class and + is now understood as the ordinal addition. Can we follow this schema to contruct interesting infinite algebraic structures?
For , we have for any infinite the relation so . Hence and the initial problem is reduced to the case where contains only finite cardinals (= finite ordinals) and so the problem is still not really exciting. What about the case ? In general it is possible to have without being equal and so for any such that , we get that . In particular for any infinite ordinal and such that we obtain because (if this is not obvious to you, a more general statement is proved below). As a consequence, more general assumptions on the class strongly limit the structure of the group. For example if is closed under an infinite sum ( for example) then is trivial. Similarly, if we require to be stable by multiplication (in order to define a ring structure for example) then is trivial whenever contains an infinite ordinal (hence the remaining case is again ). If is not trivial, we denote the least element such that (in particular ).
One additional natural hypothesis should be added. We know that for any ordinal there exists a unique ordinal such that . We would very like to match the difference "". For that purpose, we only require to belong to . Hence we assume that
Let's come back to the case with the new assumption (3) and suppose that is not trivial. Then and any finite and be written with . By (3), and because is stable by finite sums, . Thus is the monogenic group generated by .
What about the general case? Unfortunately, it turns out that if is not trivial it is still a monogenic group generated by . To prove this, we need to recall some equalities on ordinals. First, for any , we have . This is clearly true for and we prove the general case by induction on : and . Now, if we have two ordinals and if is such that we have . Finally, if we have two ordinals we can write their Cantor Normal Forms:
where and are positive integers. Using the equality for it is clear that if .
Now our element can be written where is the first term of its Cantor Normal Form. For any , we can write its euclidean division by : where . By the previous discussion, if then we can write the Cantor Normal Forms of the two elements as in (4) and see that . Hence and so , which is a contradiction. So and because is stable by finite sums, and by the property (3) we get . This means that and hence is generated by as claimed above.
As a conclusion, if satisfies the properties (1), (2) above and is a group then . If satisfies properties (1), (2), (3) above and is a group then is isomorphic to or to . Conversely, we can build these groups from by defining the relation as the equality (respectively the equality modulo ). But we do not get any new groups...