Exponentiation and Duality

In a unified treatment of cardinal and ordinal arithmetic G. Birkhoff [18], [20] defined (cardinal) exponentiation of ordered sets: for ordered sets X and Y, X Y (the power) is the set of all order-preserving maps of Y (the exponent) to X (the base) ordered componentwise. Our aim. is to review a significant body of results concerning powers of ordered sets and to present some central open problems arising in recent work. Roughly, we have two topics: first, an analysis of the structure of powers and their symmetries; second, a study of duality results for lattice-ordered algebras.

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