# A non-commutative cyclic semigroup

Allowing exponents of variables to reach every transfinite numbers we can obtain counterexamples to consolidated algebraic results.

Let be the ordered set of all transfinite numbers.

**A.Vitali’s semigroup: a non commutative cyclic semigroup:**

Consider the set . It satisfies the following properties:

1) it is a semigroup with unity, a monoid:

closure:

associativity: it comes from additive associativity of transfinite numbers

unity: is a good unity

2) it is cyclic: by the very definition every element is a power of x !!! But

3) it is not commutative; while

4) it admits right but not left cancellation law: it is a consequence of analogue additive properties of transfinite numbers

Remark: in reality itself is such an example but we have introduced x to have a multiplicative form**.**

**A.Vitali’s ring: a polinomial ring in one variable in which the powers of the variable don’t commute:**

Consider the usual polynomial ring in one indeterminate x commuting with the elements of a field k. But now we allow the exponents of x to go through all . Then we have an extended polynomial ring i in one variable x in which, by the preceeding example, the powers of the indeterminate don’t commute !!!

It contains as a subring with its usual properties.

**A.Vitali’s transfinite power series ring:**

We can also consider the ring of transfinite powers series with the usual meaning that contains as a subring the ring of infinite power series in x.

It is an easy exercise to see that here again is right inverse of (1-x):

But this is not the only right inverse: also the traditional in k[x] is, and those with all zero from the second w-segment too, and so on. A right inverse is given by the chose of a subset of these w-segments that constitute .The first w-segment, that of natural numbers must be present. The successive w-segments are indexed, by little thought, by itself. To distinguish we call it . In definitive (1-x) has at least right inverses such as , and in general .

It is easy to see that there many others: if we also permit coefficients to the powers, because of the well ordering of , it is necessary and sufficient, guided by the preceeding calculations, that all consecutive terms inside the same w-segment cancel two by two, For this it is necessary and sufficient that all coefficients inside the w-segment be equal. The first coefficient of the first w-segment must be one while the first of the others can be arbitrary. Then the most general right inverse of (1-x) is of the form: .

However: is still left inverse of (1-x), because we can think the calculation in , but is no more:

The same calculations with the most general right inverse gives as a result .

It is easy to see that is the only left inverse of (1-x). In fact dividing the transfinite powers of this left inverse in w-segments as usual we see that:

1) the first w-segment must be

2) Inside the same w.segment to cancel two by two we must have the same coefficient

3) the leading term of the w-segment survives and hence must have coefficient equal to zero.

**References
**H. Bachmann Transfinite Zahlen Springer Verlag 1967

J. Donald Monk Introduction to set theory Mc Graw Hill New York 1969

Lectures on Set Theory euclid.colorado.edu/~monkd/setth.pdf

M. Di Nasso people.dm.umipi.it/dinasso/~Mauro Di Nasso

T.Y.Lam A First Course in Noncommutative Rings Springer