Arithmetic of ordered systems

Abstract

This paper was first conceived as a short note in which two operationscalled ordered addition and ordered multiplication-were to be defined for ordered systems and shown to include all but the sixth of the assorted operations of ordinal and cardinal addition, multiplication, and exponentiation discussed by G. Birkhoff in [1 ](1). These facts are still in the paper but are completely overshadowed by far more important considerations, mostly arising from the rather unexpected properties of the operation of ordered multiplication. The general purpose of this paper is easily explained. We define these operations of ordered addition and multiplication of families of systems and define certain unary operations called transitization and contraction, which are applied to single systems. We wish to discuss, first, the properties of these operations singly and in combination, and, second, the nature of the ordered systems which arise when these are applied to systems with assigned properties. Examples of the first type of theorem are the general associative laws satisfied by ordered addition and multiplication; a sample of the second type is Theorem 5.14 which shows that while the product of transitive systems need not be transitive it has a property (defined below) which is closely allied to transitivity. The systems (called numbers) studied by Birkhoff have the two properties of transitivity (if a _ b _ c, then a _ c) and antisymmetry (if a > b > a, then a= b). It is noted in [1] that the ordinal power of such systems need not be antisymmetric; that transitivity also fails is easily seen by an example (see ?3 below) in which the base is a two-element well-ordered system and the exponent is the system of integers ordered by magnitude. It can be seen from the systems used in this example that any restriction on base and exponent so great that the ordinal power is transitive must be very strong indeed. (For example, we show in ?4 that when base and exponent are both numbers, the ordinal power is a number if and only if the base is a cardinal number or the exponent satisfies the ascending chain condition.) In this paper an ordered system T = (R, _) will be a set R in which a reflexive binary relation > holds between some pairs of elements of R. The preceding paragraph shows why no further restriction is placed on the systems involved; apparently no reasonable subclass is closed under ordinal exponentiation. Even the ordinal power of countable ordinals leads to nontransitive systems! Since we often prefer transitive systems or numbers, this