Arithmetic of Ordered Sets

Abstract

The fundamental operations of ordinal and cardinal arithmetic have been extended to arbitrary binary relations and their isomorphism types. (Whitehead and Russell [1912], pp. 291 ff.) When applied to (partially) ordered sets, these operations yield new ordered sets. A unified treatment of these operations was initiated in Birkhoff [1937] and [1942], where it was shown that most of the basic laws of arithmetic apply in this general setting. See also Birkhoff [1967], pp. 55 ff.

Many of the deeper properties of ordered addition were listed without proofs in Lindenbaum and Tarski [1926], and these properties were developed axiomatically in Tarski [1956]. More specifically, the operations treated there were ordinal addition, as applied to pairs of types and to ω-sequences of types, and conversion. The axioms for ordinal algebras hold for the isomorphism types of arbitrary binary relations, and most of the known properties of these operations are consequences of the axioms. However, some facts about ordinal addition cannot even be formulated within this framework. A notable example is Aronszajn’s [1952] characterization of commuting pairs of isomorphism types.

The lexicographic product has an important role in the development of the arithmetic of ordinal numbers, and in the study of total order (Hausdorff [1914]). For finite reflexive relations (digraphs) this operation was investigated in Dorfler and Imrich [1972].

Every relation has a unique representation as the cardinal sum (disjoint union) of connected relations, and the study of cardinal addition therefore largely reduces to the study of cardinal numbers. The most important fact about direct (or cardinal, or Cartesian) products is Hashimoto’s [1951] result that any two representations of a connected ordered set as a direct product have isomorphic refinements. From this various cancellation and unique factorization results follow. Without the assumption of connectedness, the unique factorization property fails, even for finite ordered sets, but the cancellation property holds for arbitrary finite relations with a reflexive element (Lovász [1967]).

The (cardinal) power A ^B , where A and B are ordered sets, is defined (Birkhoff [1937]), to be the set of all isotone maps from B to A with the order inherited from the direct power A ^∣B∣ As Birkhoff noted, this operation satisfies the usual laws for exponents. More difficult problems arise when we consider conditions under which the cancellation laws for the bases and for exponents hold:

$${A^C} \simeq {A^D}{\text{ implies }}C \simeq D,$$

$${A^C} \simeq {B^C}{\text{ implies }}A \simeq B,$$

and conditions which insure that two representations

$$P \simeq {A^C}{\text{ and }}P \simeq {B^D}$$

have common refinements

$$A \simeq {E^X},B \simeq {E^Y},C \simeq YZ,D \simeq XZ.$$

These problems were investigated in Bergman, McKenzie and Nagý [a], Duffus [1978], Duffus, Jónsson and Rival [1978], Duffus and Rival [1978], Duffus and Wille [1979], Fuchs [1965], Jonsson and McKenzie [a], Novotný [1960], and Wille [1980].

This work was supported by NSF Grant MCS 7901735.