Kimmo I. Rosenthal

The theory of quantaloids

Quantaoids new quantaloids from old free quantaloids and Q-enriched categories an example - a categorical approach to automata and tree automata modules, bimodules and *-autonomous categories.

Spatial sublocales and essential primes

Abstract In this paper, we characterize those locales L such that every sublocale is spatial. The notion of essential prime is introduced and it is shown that every sublocale of L is spatial iff every element of L is the meet of its essential primes. This is related to work of Simmons characterizing when the assembly NL of nuclei on L is spatial. If L is the open subsets of a space X , these conditions correspond to X being weakly scattered. Finally, the main theorem is applied to the case X = spec R , where R is a commutative ring with identity.

Autonomous categories of bimodules

Linear logic, a logical system developed by Girard [13] is a logic of resources which has elicited much interest from theoretical computer scientists because of its numerous potential applications. (See [36] for a brief overview and also [38].) It has also drawn the interest of logicians and category theorists. The connection with category theory comes about from the fact that the notion of *-autonomous category, due to Barr [2], provides a categorical model for a significant portion of linear logic. (See also his more recent exposition [3], as well as the work of Seely [37] and Blute [6,7]). Thus, attempts to find mathematical models for various aspects of linear logic center around *-autonomous categories and there have been interesting recent developments concerning connections with the theory of Petri nets [24,25]. Also, the notion of weakly distributive categories [l l] has been developed to model aspects of linear logic. The partially ordered (complete) models are called Girard quantales and they were first extensively studied by Yetter [40] and then by Rosenthal [31] (also see Chapter 6 in [30]). There are several interesting non-commutative Girard quantales, such as Rel(X), the relations on a set X, and Ord(P), the order ideals on a preordered set P. Until now, there has not been much in the way of examples of non-symmetric *-autonomous categories (other than partially ordered ones) as potential models for non-commutative linear logic. Recently, Barr [4] developed a non-symmetric version of the Chu construction for *-autonomous categories [2], and Blute has obtained non-symmetric *-autonomous categories by considering quantum groups (quasitriangular Hopf algebras) [S]. In this paper, we describe a-general way of constructing a special class of nonsymmetric *-autonomous categories, which we call cyclic, from a given *-autonomous category 9, by using enriched category theory and the calculus of 5?-bimodules. If Y is an autonomous category, we make the observation, following Lawvere [23], that

Strong de Morgan's law and the spectrum of a commutative ring

The logical principle (A G-B) V (B +A) = true, known as the Strong de Morgan’s law, is not in general valid in intuitionistic logic. P. T. Johnstone (in [6]) showed that this principle holds in the topos sh(X), of set-valued sheaves on a topological space X, and hence also in the locale O(X) of open subsets of X, if and only if every closed subspace of X is extremally disconnected. We investigate this property for X = Spec R, the spectrum of a commutative ring R with identity, and obtain ideal theoretic conditions characterizing those R whose spectra satisfy the Strong de Morgan’s law. These ideal theoretic properties are closely related to ones which characterize Dedekind domains, however, they involve the consideration of radical ideals. Section one develops the notion of a closed poset, which is a closed category whose underlying category is a partially ordered set. The main examples of closed posets that we consider are locales and ideals of a commutative ring R. We carry the analogy between the two examples further by establishing an identification between the locale B(Spec R) and the locale RIdl(R) of radical ideals of R. Section two presents the Strong de Morgan’s law and Johnstone’s results about the de Morgan laws for sh(X) and B(X). Using the analogy developed in section one, we define algebraic de Morgan’s laws for rings. This leads directly to our main theorem, which gives equivalent ideal theoretic conditions characterizing rings R, such that Spec R satisfies the Strong de Morgan’s law. If Spec R is Noetherian, we obtain several additional equivalences.

Quantaloids, enriched categories and automata theory

This article is intended to be an survey article outlining how the theory of quantaloids and categories enriched in them provides an effective means of analyzing both automata and tree automata. The emphasis is on the unification of concepts and how categorical methods provide insight into various calculations and theorems, both illuminating the original presentation as well as yielding conceptually simpler proofs. Proofs will be omitted and the emphasis is on providing the reader (even a relatively inexperienced one) with an understanding of the basic constructions and results.

Local equivalence relations

Abstract In this paper, we investigate the concept of local equivalence relation, a notion suggested by Grothendieck. A local equivalence relation on a topological space X is a global section of the sheaf of germs of equivalence relations on X. We investigate the extent to which a local equivalence relation can be described by a global one and analogously when can a global equivalence relation be recovered from its associated local one. We also look at the notion of a fiber map, which sheds further light on these concepts. A motivating example is that of a foliation on a manifold.

Quantaloidal nuclei, the syntactic congruence and tree automata

In [3], Betti and Kasangian indicate how one can approach the study of tree automata from the perspective of enriched category theory. If & is an algebraic theory (in the sense of Lawvere [13]), then one can construct a bicategory g(d) so that a tree automaton can be realized as a category enriched in g)(d) subject to certain conditions, together with an initial and final bimodule. The composition of these bimodules results in the forest (set of trees), which is called the behavior of the automaton. Kasangian and Rosebrugh have pursued this idea further in subsequent papers [ 10,111. The bicategory g’(d) is an example of a quantaloid. A quantaloid is a category enriched in the category XY of sup-lattices and it is a natural generalization of the notion of a quantale. Quantales include frames, lattices of ideals, and relations as examples and their basic theory and some applications have been collected together in [18]. Quantaloids were studied by Pitts [17] in investigating distributive categories of relations and topos theory. Some of the basic features of quantaloids, viewed as generalized quantales, including a discussion of enriched categories, were developed by Rosenthal in [20] and some further aspects were examined in [21]. In this paper, we propose using the theory of quantaloidal nuclei (which generalize the notion of a nucleus on a quantale [18] or a frame [9]) to study syntactic congruences, which arise in the theory of automata and tree automata. If 5! is a quantaloid and f : c-+ d is a morphism of 9, one can construct a

Sheaves of integral domains on stone spaces

It is well known in the theory of ring representations on Stone spaces, that if R is a commutative ring with 1, it is representable as a sheaf of fields (local rings) on a Stone space iff R is a von Neumann regular (exchange) ring. These results make use of the Pierce representation of R. The question of necessary and sufficient conditions on R to guarantee representability as a sheaf of integral domains is answered in this article. The appropriate condition on R is that of being an ‘almost weak Baer’ ring, where this means that Ann(a) is generated by its idempotent elements for all a e R. Two examples from rings of continuous functions distinguish this property from several closely related ring theoretic conditions.

Ideals of closed categories

A complete closed poset can be viewed as a commutative monoid in the closed category SI of complete lattices and sup-preserving maps. A lax adjunction between closed posets and the 2-category CSI of symmetric, monoidal closed categories over sup-lattices is described. This makes use of categories of ‘modules’ over a closed poset. If V is a suitably complete and cocomplete symmetric monoidal closed category, it is shown that the subobjects of the unit for ⊗ in V form a closed poset. The functoriality of this ideal construction is investigated; it is functorial in two different ways depending upon the type of morphism we consider for our closed categories. It alternately provides a right lax or right colax adjoint to the inclusion of closed posets into the 2-category of closed categories under consideration.

Corrigendum: A completion for partially ordered abelian groups

I t has come to my a t ten t ion that the proof of Theorem 2.8 in my a r t i c l e (Semigroup Forum Vol. 28 (1984), 273-290) is not cor rec t I t is claimed that i f (W,H) is a semicontinuous closed poser and i f G ~H is a Scott continuous homomorphism of p a r t i a l l y ordered abelian groups, then i f w ~ W, the set 7(w) = {a C GIf (a) < w} is Scott closed in G. This need not be t rue, i f sups in H are not preserved when computed in W. The proof can be amended by changing the d e f i n i t i o n of semicontinuous closed poset as fo l lows. A semicontinuous closed poset is a pai r (V,G) sa t i s f y ing De f i n i t i on 2.6 (pg. 284) together with the requirement that the inc lus ion G:-~ V preserve d i rected sups. This addi t ional assumpt ion does not a f f ec t the primary example considered in the paper.