When is a natural duality ‘good’?

Building on the most current work in the theory of natural dualities, we continue the study of strong dualities for the quasi-variety generated by a finite algebra. We investigate ten different versions of what we would like to mean by a good duality. Each version concerns, among other things, a specific restriction on the type of the structures in the dual category which insures that the dual structures will in a useful sense be simple. Through each investigation we seek a theorem characterizing, in terms of finitely verifiable conditions, those finite algebras generating a quasi-variety which admits a strong duality meeting the given restrictions. Our study includes a careful treatment of coproducts, logarithmic dualities and strong dualities by various unary structures.Dedicated to the memory of Alan DayPresented by J. Sichler.Research supported by a 1992 ARC Grant (Davey).     

Pathwise Duals of Monotone and Additive Markov Processes

This paper develops a systematic treatment of monotonicity-based pathwise dualities for Markov processes taking values in partially ordered sets. We show that every Markov process that takes values in a finite partially ordered set and whose generator can be represented in monotone maps has a pathwise dual process. In the special setting of attractive spin systems, this has been discovered earlier by Gray. We show that the dual simplifies a lot when the state space is a lattice (in the order-theoretic meaning of the word) and all monotone maps satisfy an additivity condition. This leads to a unified treatment of several well-known dualities, including Siegmund’s dual for processes with a totally ordered state space, duality of additive spin systems, and a duality due to Krone for the two-stage contact process, and allows for the construction of new dualities as well. We show that the well-known representation of additive spin systems in terms of open paths in a graphical representation can be generalized to additive Markov processes taking values in general lattices, but for the process and its dual to be representable on the same underlying space, we need to assume that the lattice is distributive. In the final section, we show how our results can be generalized from finite state spaces to interacting particle systems with finite local state spaces.     

New-from-old full dualities via axiomatisation

We study different full dualities based on the same finite algebra. Our main theorem gives conditions on two different alter egos of a finite algebra under which, if one yields a full duality, then the other does too. We use this theorem to obtain a better understanding of several important examples from the theory of natural dualities. We also clarify what it means for two full dualities based on the same finite algebra to be different. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual categories.     

Full but not strong dualities at the finite level: Extending the realm

The realm of natural dualities that are known to be full but not strong at the finite level is a very small one, consisting of a single example. This example, based on the three-element bounded distributive lattice, was presented by Davey, Haviar and Willard [8]. In this paper, we extend this realm to the class of all natural dualities based on a finite non-boolean bounded distributive lattice. Received June 15, 2005; accepted in final form November 26, 2005.     

Duality in quasi-convex supremization and reverse convex infimization via abstract convex analysis,and applications to approximation **

We give duality theorems and dual characterizations of optimal solutions for abstract quasi-convex supremization problems and infimization problems with abstract reverse convex constraint sets. Our main tools are dualities between families of subsets, conjugations of type Lau associated to them, and subdifferentials with respect to conjugations of type Lau. These tools permit us to give explicitly the relation between.the constraint sets, and the relation between the objective functions, of the primal problem and the dual problem. As applications, we obtain duality theorems for quasi-convex supremization and reverse convex infimization in locally convex spaces and, in particular, for worst and best approximation in normed linear spaces.     

Using coloured ordered sets to study finite-level full dualities

We consider all the full dualities for the class of finite bounded distributive lattices that are based on the three-element chain 3. Under a natural quasi-order, these full dualities form a doubly algebraic lattice ${\mathcal{F}_{\underline{3}}}$ . Using Priestley duality, we establish a correspondence between the elements of ${\mathcal{F}_{\underline{3}}}$ and special enriched ordered sets, which we call ‘coloured ordered sets’. We can then use combinatorial arguments to show that the lattice ${\mathcal{F}_{\underline{3}}}$ has cardinality ${2^{\aleph_{0}}}$ and is non-modular. This is the first investigation into the structure of an infinite lattice of finite-level full dualities.     

Are dualities appropriate for duality theories in optimization?

We raise some questions about duality theories in global optimization. The main one concerns the possibility to extend the use of conjugacies to general dualities for studying dual optimization problems. In fact, we examine whether dualities are the most general concepts to get duality results. We also consider the passage from a Lagrangian approach to a perturbational approach and the reverse passage in the framework of general dualities. Since a notion of subdifferential can be defined for any duality, it is natural to examine whether the familiar interpretation of multipliers as generalized derivatives of the performance function associated with a dualizing parameterization of the given problem is still valid in the general framework of dualities.     

Finite type 3-manifold invariants and the structure of the Torelli group. I

Using the recently developed theory of finite type invariants of integral homology 3-spheres we study the structure of the Torelli group of a closed surface. Explicitly, we construct (a) natural cocycles of the Torelli group (with coefficients in a space of trivalent graphs) and cohomology classes of the abelianized Torelli group; (b) group homomorphisms that detect (rationally) the nontriviality of the lower central series of the Torelli group. Our results are motivated by the appearance of trivalent graphs in topology and in representation theory and the dual role played by the Casson invariant in the theory of finite type invariants of integral homology 3-spheres and in Morita's study [Mo2, Mo3] of the structure of the Torelli group. Our results generalize those of S. Morita [Mo2, Mo3] and complement the recent calculation, due to R. Hain [Ha2], of the I-adic completion of the rational group ring of the Torelli group. We also give analogous results for two other subgroups of the mapping class group. Oblatum 19-IX-1996 & 13-V-1997     

Enumeration of non-positive planar trivalent graphs

In this paper we construct inverse bijections between two sequences of finite sets. One sequence is defined by planar diagrams and the other by lattice walks. In [13] it is shown that the number of elements in these two sets are equal. This problem and the methods we use are motivated by the representation theory of the exceptional simple Lie algebra G ₂. However in this account we have emphasised the combinatorics.     

On Finite Maximal Antichains in the Homomorphism Order

We show that for structures with at most two relations all finite maximal antichains in the homomorphism order correspond to finite homomorphism dualities. We also show that most finite maximal antichains in this order split.     

A Characterization of Self-similar Symbolic Spaces

In this paper we use fractal structures to study self-similar sets and self-similar symbolic spaces. We show that these spaces have a natural fractal structure, justifying the name of fractal structure, and we characterize self-similar symbolic spaces in terms of fractal structures. We prove that self-similar symbolic spaces can be characterized in a similar way, in the form, to the definition of classical self-similar sets by means of iterated function systems. We also study when a self-similar symbolic space is a self-similar set. Finally, we study relations between fractal structures with “pieces” homeomorphic to the space and different concepts of self-homeomorphic spaces. Along the paper, we propose several methods in order to construct self-similar sets and self-similar symbolic spaces from a geometrical approach. This allows to construct these kind of spaces in a very easy way.     

Morse Novikov theory and cohomology with forward supports

We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson [HL]. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward' dualities are also established: a Lambda duality over the Novikov Ring and a Topological Vector Space duality over the reals.     

Non-invertible-element constacyclic codes over finite PIRs

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities.     

Boolean Topological Distributive Lattices and Canonical Extensions

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. The second author was supported by Slovak grants VEGA 1/3026/06 and APVV-51-009605.     

Comparison between different duals in multiobjective fractional programming

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.      

Piggyback dualities revisited

In natural duality theory, the piggybacking technique is a valuable tool for constructing dualities. As originally devised by Davey and Werner, and extended by Davey and Priestley, it can be applied to finitely generated quasivarieties of algebras having term-reducts in a quasivariety for which a well-behaved natural duality is already available. This paper presents a comprehensive study of the method in a much wider setting: piggyback duality theorems are obtained for suitable prevarieties of structures. For the first time, and within this extended framework, piggybacking is used to derive theorems giving criteria for establishing strong dualities and two-forone dualities. The general theorems specialise in particular to the familiar situation in which we piggyback on Priestley duality for distributive lattices or Hofmann–Mislove– Stralka duality for semilattices, and many well-known dualities are thereby subsumed. A selection of new dualities is also presented.     

The dual geometry of Boolean semirings

It is well known that the variety of Boolean semirings, which is generated by the three element semiring ${\mathbb{S}}$ , is dual to the category of partially Stone spaces. We place this duality in the context of natural dualities. We begin by introducing a topological structure S? and obtain an optimal natural duality between the quasi-variety ISP( ${\mathbb{S}}$ ) and the category IS _c P+(S?). Then we construct an optimal and very small structure S? _os that yields a strong duality. The geometry of some of the partially Stone spaces that take part in these dualities is presented, and we call them “hairy cubes”, as they are n-dimensional cubes with unique incomparable covers for each element of the cube. We also obtain a polynomial representation for the elements of the hairy cube.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).