Some iterated natural sums

We study a transfinite generalization of the ordinal (Hessenberg) natural obtained by taking suprema at limit stages. We show that a transfinite natural differs from the more usual transfinite ordinal only for a finite number of iteration steps. We show that the transfinite natural of a sequence of ordinals can be obtained as a sum (in an order-theoretical sense) of those ordinals, in fact, it is the largest mixed which satisfies a finiteness condition, relative to the ordering of the sequence.