We study general lattices with normal unary operators for which we prove relational representation and duality results. Similar results have appeared in print, using Urquhart’s lattice representation, by the second author with Vakarelov, Radzikowska and Rewitzky. We base our approach in this article on the Hartonas and Dunn lattice duality, proven by Gehrke and Harding to deliver a canonical lattice extension, and on recent results by the first author on the relational representation of normal lattice operators. We verify that the operators at the representation level (appropriately generated by relations) are the canonical extensions of the lattice operators, in Gehrke and Harding’s sense.

It has been argued that in the context of automated theorem proving deduction procedures based on a frame semantics is more efficient than those based on algebraic semantics, for some logics. Frame semantics, for several logics, is specified by means of a representation and Stone-type duality result, involving a topology on the frame which, however, is not relevant in proving soundness of the logic. This has led to the development of a research program on Discrete Dualities, where a number of relevant results have been published over the past decade, but the program seems to have stumbled on the case of frames for non-distributive logics and bounded lattices with operators. In this article, we fill in this gap by presenting discrete dualities for bounded lattices and for lattices with one-place modal operators. Our results are extendible to discrete dualities for any normal lattice expansion.

With the development of the Web and the intense research for a semantic Web, the need arises for standardized and rigorous semantic specification both of services on the semantic Web and of the objects and processes that these services offer to a potential client. In this report, our focus is with learning services, i.e. Web services whose purpose is to offer educational services to clients. Such services are the owners of learning objects, which they deliver for use to their clients. Though the need for standardizing learning objects has become apparent, there is no substantial contribution towards a reference architecture for learning objects. We introduce here the CROP architecture, based on the notions of concept, resource, order and product.