Ordinal semigroups

Abstract

In a previous paper we introduced a version of associativity for a partial infinitary operation. We prove here that if ${\hat{I}}^3$ is an infinite ordinal and some associative infinitary operation is defined for all sequences indexed by ordinals $\le {\hat{I}}^3$, then such an operation can be uniquely expanded to apply to every sequence indexed by any ordinal of cardinality $|{\hat{I}}^3|$. In particular, if some associative operation is defined for all finite sequences as well as for all $\ddot{I}‰$-indexed sequences, then the operation can be uniquely expanded to apply to every sequence indexed by a countable ordinal.

11 pages