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Exponentiation and Duality

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Ordered Sets

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 83))

Abstract

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.

Birkhoff [18] observed that exponentiation of ordered sets satisfies the usual laws for exponents and raised questions concerning cancellation laws for bases and exponents [20]: Does either X Y≅X Z or Y X≅Z X imply Y≅Z? Early results appear in M. Novotný [59] and E. Fuchs [40]• A logarithmic property is used by D. Duffus and I. Rival [37] to obtain cancellation of bases for finite ordered sets. Using extensions of this method and the relation between symmetries of X Y and both product and exponential decompositions of X Y, B. Jónsson and R. McKenzie [52] obtained many cancellation and refinement theorems. For instance, under conditions on ordered sets A, B, C, D (too varied and technical to state here) they show that A C≅B C implies A≅B, and that A C≅B D yields E, X, Y, Z and isomorphisms A≅E X, B≅E Y, C≅Y·Z, D≅X·Z. (See [16], [36], [39], [87] for related results.)

It remains to be seen if, in the finite case, cancellation of exponents is always possible. Other open problems concern simplification of conditions insuring refinement properties and unification of existing results (cf. [52]).

Concerning symmetries of powers, it is easy to see that for ordered sets X and Y, Aut(X) × Aut(Y) is embeddable in Auti(X Y). In [52] two questions are raised: under what conditions is the embedding full, and, if not full, can Aut (X Y) still be described in terms of Aut(X) and Aut(Y)? Jónsson and McKenzie [52] obtained some results: Aut(X Y) ≅ Aut(X) × Aut(Y) if X is a subdirectly irreducible lattice and Y is finite and connected. Duffus and Wille [38] provided answers to both questions when X is a lattice of finite length. Most recently, Jónsson [51], building on techniques of [52], has obtained much stronger results.

Birkhoff [19] proved that every finite distributive lattice D is isomorphic to \( {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2}}}^{{J{{{(D)}}^{d}}}}}where {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2}} \) is the two-element chain and J(D)d is the dual of the ordered subset of join-irreducible elements of D. Implicit in this is the fact that the category \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{D} _F}\) of finite bounded distributive lattices is dually equivalent to the category \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} _F}\) of finite ordered sets. This duality was extended to infinite bounded distributive lattices by H.A. Priestley [61], [62]: the category \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{D} \) of all bounded distributive lattices is dually equivalent to the category \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \) of all compact Hausdorff ordered spaces which are embeddable in a power of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2} \).

Further extension of this duality is possible. A finite lattice L is order-polynomial-complete if for every positive integer n, every order-preserving map of L n to L is an n-ary polynomial (alias, algebraic) function [70], [86], [87]. For instance, Wille [85] showed that every finite simple lattice whose greatest element is the join of atoms is order-polynomial-complete. A proof of Priestley’s duality result utilizing the order-polynomial-completeness of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2} \) is given in B.A. Davey [28]. The implicit possibility that a similar duality holds for each order-polynomial-complete lattice was exploited in B.A. Davey, D. Duffus, R.W. Quackenbush and I. Rival [30J and further refined in B.A. Davey and I. Rival [32].

The dual equivalence of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{D} \) (respectively, \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{D} _F}\)) and \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \) (respectively, \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} _F}\)) allows representation of both the objects and the morphisms of D in terms of those of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \). Moreover the objects and morphisms of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \), especially of \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} _F}\), are so pictorial that this duality is a very powerful tool for the study of particular bounded distributive lattices and the class they comprise, and for generating examples and counterexamples. Further, this duality reveals interplay between ordered sets and distributive-lattice-ordered algebras — algebraic structures often arising in logic such as Boolean algebras, Heyting algebras, and pseudo-complemented distributive lattices. Provided that one can describe those ordered spaces in \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \) which arise as duals of these enriched structures and the continuous order-preserving maps in \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{P} \) which are dual to their homomorphisms, the Birkhoff-Priestley result immediately yields a duality for the class of enriched structures. In this setting, natural algebraic questions can lead to order-theoretic considerations which are interesting in their own right. (B.A. Davey [28] and B.A. Davey and H. Werner [33] provide a general approach to duality theory.)

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Authors and Affiliations

  1. Department of Pure Mathematics, La Trobe University, Victoria, 3083, Australia

    B. A. Davey

  2. Department of Mathematics, Emory University, Atlanta, Georgia, 30322, USA

    D. Duffus

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  1. B. A. Davey
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  2. D. Duffus
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  1. Department of Mathematics and Statistics, The University of Calgary, Canada

    Ivan Rival

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Davey, B.A., Duffus, D. (1982). Exponentiation and Duality. In: Rival, I. (eds) Ordered Sets. NATO Advanced Study Institutes Series, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7798-3_2

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