Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 (IRL0) with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c (the underlying set of CM is the set of elements greater or equal than c). If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, CM is isomorphic to CeM , and Ce is adjoint to ( ), where ( ) assigns to an object A of IRL0 the product A × A 0 which is an object of NRL.

Link to this article: http://journals.cambridge.org/abstract_S0960129502003699How to cite this article: MATÍAS MENNI and ALEX SIMPSON (2002). Topological and limit-space subcategories of countably-based equilogical spaces.There are two main approaches to obtaining 'topological' cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed -for example, the category of sequential spaces. Under the other, one generalises the notion of space -for example, to Scott's notion of equilogical space. In this paper, we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. In fact, this category turns out to be equivalent to the category of all quotient spaces of countably based topological spaces. We show that the category is bicartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the category of equilogical spaces. We also show that the category of countably based equilogical spaces has a larger full subcategory that can be simultaneously viewed as a full subcategory of limit spaces. This full subcategory is locally cartesian closed and the embeddings into limit spaces and countably based equilogical spaces preserve this structure. We observe that it seems essential to go beyond the realm of topological spaces to achieve this result.Of course very many reconciliations of this situation have been proposed. One possibility is to cut down the category of topological spaces to a full subcategory that is cartesian closed. Some well-known examples are: Steenrod's category of compactlygenerated Hausdorff spaces (Mac Lane 1971); the category, Seq, of sequential spaces (which contains many computationally important non-Hausdorff spaces) (Hyland 1979b); or the even larger category of quotients of exponentiable spaces considered in Day (1972). However, the received wisdom about such categories is that their function spaces are topologically hard to understand. It is much quoted that the exponential N B , where B is Baire space, can never be first-countable (Hyland 1979b), whereas an ideal approach from a computational viewpoint would allow effectivity issues to be addressed, and the stricter requirement of second-countability is often claimed to be necessary for such, see, for example, Smyth (1992).A second alternative is to expand the category Top by adding new objects and hence new potential exponentials. Again there are many ways of doing this. A very elegant construction is to take the regular completion of Top (as a left-exact category) or the related exact completion Carboni and Rosolini 2000;Rosolini 2000). The regular completion has a straightforward description as a category of equivalence relations on topological spaces, whose importance (in...

No abstract

We prove that every integral rig in Set is (functorially) the rig of global sections of a sheaf of really local integral rigs. We also show that this representation result may be lifted to residuated integral rigs and then restricted to varieties of these. In particular, as a corollary, we obtain a representation theorem for pre-linear residuated join-semilattices in terms of totally ordered fibers. The restriction of this result to the level of MV-algebras coincides with the Dubuc-Poveda representation theorem.