Arithmetic of Ordered Sets

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.

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