  This is the first in a series of several articles. Our general purpose is to
investigate Carlson's <_1-relation in the whole class of ordinals and later
link it with ordinals of proof-theoretic interests. In this introductory
article, after giving the basic definitions, we study the (canonical)
isomorphisms {g(0,a,b)|a,b are additive principal numbers} and the <^0-relation
and see how it is that <_1 induces, through all of these notions, thinner
k-club classes of ordinals. In coming articles it will be shown the complete
generalization of these ideas to the thinnest k-club classes induced by <_1.
