Regressive order-types.
John L. Hickman · Notre Dame Journal of Formal Logic · 1977
It is of course well-known that with respect to addition, ordinals possess the property of left-cancellation, but that this property is not shared by order-types in general.In this note* we introduce the property of regressiveness, show that an order-type is regressive if and only if it cannot be additively left-cancelled in general, and give a simple canonical form for regressive order-types.We conclude by giving criteria for the regressiveness of an order-type possessing a (nontrivial) ordinal right divisor.Lower-case Greek letters are used to denote order-types, and uppercase Latin letters to denote (ordered) sets, and we usually suppress mention of the order relation on a set.A is called a "representative" set for a if a is the isomorphism type of A. The ordered union and ordered product of sets A, B are respectively denoted by U A + B" and (( A x B".If A = B + C +D, then B{D) is called an "initial (final) segment" of A, and C is sometimes called an "interval" of A. The same terminology is used for order-types, although we sometimes refer to a final segment of an order-type as a remainder.A (strict) order-preserving map / : A -* B is called an "isomorphism", and if B =f"A, then we write "/ : A ^ B".The first transfinite ordinal is denoted by "ω": in general we use "σ", "τ", "p" for ordinals and "α", "β", "y", ... for general ordertypes, "i", "j", "k", . . .are used to denote finite ordinals.The converse order-type of a is denoted by "α*".An elementary property of ordinals is that of left-cancellation: given ordinals σ, r, p, if σ + r = σ + p, then τ = p.This property is not shared by all order-types, since for example if η is the order-type of the rationale under their usual ordering, we have.