Natural dualities between abelian categories

In this paper we consider a pair of right adjoint contravariant functors between abelian categories and describe a family of dualities induced by them.     

Binary homomorphisms and natural dualities

We investigate ways in which certain binary homomorphisms of a finite algebra can guarantee its dualisability. Of particular interest are those binary homomorphisms which are lattice, flat-semilattice or group operations. We prove that a finite algebra which has a pair of lattice operations amongst its binary homomorphisms is dualisable. As an application of this result, we find that every finite unary algebra can be embedded into a dualisable algebra. We develop some general tools which we use to prove the dualisability of a large number of unary algebras. For example, we show that the endomorphisms of a finite cyclic group are the operations of a dualisable unary algebra.     

Subdifferentials with respect to dualities

LetX andW be two sets and: ¯R^X ¯R^W a duality (i.e., a mapping such that for all and all index setsI). We introduce and study the subdifferential of a function at a pointx _o X, with respect to. We also consider the particular cases when is a (Fenchel-Moreau) conjugation, or a -duality, or a -duality, in the sense of [8].     

Natural dualities for semilattice-based algebras

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates. Presented by R. Willard.     

On dualities between function spaces

By [6], the dualities between and , whereX andW are two sets and (i.e., the mappings satisfying for all and all index setsI), can be represented with the aid of functions . Here we show that they can be also represented with the aid of functions , whereR = (–, +). As an application, we show that every duality is completely determined by a suitable duality between 2 ^{X ×R} and 2 ^{W ×R} (i.e., a mapping 2 ^{X ×R} 2 ^{W ×R} satisfying for all {M _i} _iI 2 ^{X ×R} and all index setsI), applied to the epigraphs of the functions .     

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.     

Infimal generators and dualities between complete lattices

Summary We give some applications of infimal generators of complete lattices, in the sense of Kutateladze and Rubinov [12], to the study of dualities between two complete lattices E and F (i.e., mappings : E F satisfying for all and all index sets I, including the empty set I=Ø). We give some additional results for E=(2^X, ), F=(2^W, ) and E=(¯R^X, ), F=(¯R^W, ) (where X and W are arbitrary sets), with suitable families of infimal generators. We obtain some lattice- theoretic properties of the relations of [22] between dualities : (2^X, ) (2^W, ), binary relations X×W, polarities (): (2^X, ) (2^W, ), coupling functionals : X×W ¯R and Fenchel-Moreau conjugations c():(¯R^X, ) (¯R^W,).     

Natural dualities for varieties of BL-algebras

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes. The preliminary research for this paper was carried out while the second author was visiting Salerno University. The second author would like to thank the first author and Salerno University for their hospitality. The second author acknowledges partial supports from Salerno University and from the belgian Fonds National de la Recherche Scientifique.     

Lightcone dualities for hypersurfaces in the sphere

In this paper, we consider hypersurfaces in the unit lightlike sphere. The unit sphere can be canonically embedded in the lightcone and de Sitter space in Minkowski space. We investigate these hypersurfaces in the framework of the theory of Legendrian dualities between pseudo‐spheres in Minkowski space.     

A unified interpretation of several combinatorial dualities

Several combinatorial structures exhibit a duality relation that yields interesting theorems, and, sometimes, useful explanations or interpretations of results that do not concern duality explicitly. We present a common characterization of the duality relations associated with matroids, clutters (Sperner families), oriented matroids, and weakly oriented matroids. The same conditions characterize the orthogonality relation on certain families of vector spaces. This leads to a notion of abstract duality.     

Good dualities and strongly quasi-injective modules

Sunto Siano R ed A due anelli e sia_RK_A un bimodulo topologico di Hausdorff su R ed A dotati della topologia discrete. Sia D(K_A) la categoria formata dagli A-moduli astratti M tali che Hom_A (M, K) separa i punti di M e sia C(_RK) la categoria formata dagli R-moduli topologici che sono topologicamente isomorfi a sottomoduli chiusi di prodotti topologici di copie di_RK. Nella prima parte del lavoro si determinano dette condizioni su K affinchè le suddette categorie siano l'una la duale dell'altra. Nella ricerca di tali condizioni la nozione di modulo topologico fortemente quasi-iniettivo giuoca un ruolo fondamentale. Nella seconda parte del lavoro si studia il caso in cui K è compatto. In particolare, si mette in evidenza uno stretto legame fra la suddetta teoria di dualità ed i risultati di K. R. Fuller sulla teoria delle equivalenze. Nella terza parte del lavoro, infine, si esamina il caso in cui_RK è disereto. Si approfondiscono in tale ambito alcuni risultati ottenuti precedentemente dagli autori nel caso in cui R ed A siano commutativi. Come applicazione si dimostrano rapidamente alcuni teoremi di B. Mütter riguardanti la dualità di Morita.

---

---

Lavoro eseguito nell'ambito dell'attività dei gruppi di ricerca matematica del CNR.     

Collineations and dualities of partial geometries

In this paper, we first prove some general results on the number of fixed points of collineations of finite partial geometries, and on the number of absolute points of dualities of partial geometries. In the second part of the paper, we establish the number of isomorphism classes of partial geometries arising from a Thas maximal arc constructed from a (finite) Suzuki-Tits ovoid in a classical projective plane. We also determine the full automorphism group of these structures, and show that every partial geometry arising from any Thas maximal arc is self-dual.     

Additive G-codes over Fq and their dualities

We define additive G-codes over finite fields. We prove that if C is an additive G-code over ${\mathbb{F}}_q$ with duality M then its dual with respect to this duality $C^M$ is an additive G-code. We prove that if M and $M^{\prime }$ are two dualities, then $C^M$ and $C^{M^{\prime }}$ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relate them to formally self-dual and linear self-dual codes.