Abstract. We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the "Full vs Strong" Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that although there is a Duality Compactness Theorem, which says that if an alter ego of finite type yields a duality at the finite level then it yields a duality, there cannot be a corresponding Full Duality Compactness Theorem. M ∼ ) does it follow that M ∼ is injective in X?This question stems from a fundamental asymmetry in all known full dualities. For every full duality between A := ISP(M) and X := IS c P + (M ∼ ), the injectivity of the algebra M in A implies the injectivity of the alter ego M ∼ in X. But the converse statement is false. Nevertheless, M ∼ is injective in X in every known example of a full duality. (The interconnections between the injectivity of M and the injectivity of M ∼ were discussed at length in [13]: see Proposition 1.11 on page 128 and pages 258-263 in the Appendix. The setting there was a general category-theoretic one and not specific to natural dualities. Some refinements in the setting of natural dualities are given in Exercises 6.2-6.6 of Clark and Davey [2].) The monograph [2] is recommended as the best source of basic facts as well as recent developments in the theory of natural dualities. Clark and Krauss [4] introduced the important notions of term-closed subsets of M S and hom-closed subsets of M S and proved that they are the same. They
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We consider properties of the graphs that arise as duals of bounded lattices in Ploščica's representation via maximal partial maps into the two-element set. We introduce TiRS graphs, which abstract those duals of bounded lattices. We demonstrate their one-to-one correspondence with so-called TiRS frames, which are a subclass of the class of RS frames introduced by Gehrke to represent perfect lattices. This yields a dual representation of finite lattices via finite TiRS frames, or equivalently finite TiRS graphs, which generalises the well-known Birkhoff dual representation of finite distributive lattices via finite posets. By using both Ploščica's and Gehrke's representations in tandem, we present a new construction of the canonical extension of a bounded lattice. We present two open problems that will be of interest to researchers working in this area.
Abstract. 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-MisloveStralka duality for semilattices, and many well-known dualities are thereby subsumed. A selection of new dualities is also presented.
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