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A Logarithmic Property for Exponents of Partially Ordered Sets

In an effort to unify the arithmetic of cardinal and ordinal numbers, Garrett Birkhoff [2; 3; 4; 5] (cf. [6]) defined several operations on partially ordered sets of which at least one, (cardinal) exponentiation, is of considerable independent interest: for partially ordered sets P and Q let PQ denote the set of all order-preserving maps of Q to P partially ordered by f ≦ g if and only if f(x) ≦ g(x) for each x ∈ Q.