Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces

The powerset monad on the category of sets does not distribute over itself. Nevertheless a weaker form of distributive law of the powerset monad over itself exists and it essentially stems from the canonical Egli-Milner extension of the powerset to the category of relations. On the other hand, any regular category yields a category of relations, and some regular categories also possess a powerset-like monad, as is the Vietoris monad on compact Hausdorff spaces. We derive the Egli-Milner extension in three different frameworks : sets, toposes, and compact Hausdorff spaces. We prove that it corresponds to a monotone weak distributive law in each case by showing that the multiplication extends to relations but the unit does not. We provide an application to coalgebraic determinization of alternating automata

Learning Automata and Transducers: A Categorical Approach

In this paper, we present a categorical approach to learning automata over words, in the sense of the L*-algorithm of Angluin. This yields a new generic L*-like algorithm which can be instantiated for learning deterministic automata, automata weighted over fields, as well as subsequential transducers. The generic nature of our algorithm is obtained by adopting an approach in which automata are simply functors from a particular category representing words to a "computation category". We establish that the sufficient properties for yielding the existence of minimal automata (that were disclosed in a previous paper), in combination with some additional hypotheses relative to termination, ensure the correctness of our generic algorithm

Lúpus comedônico: rara apresentação do lúpus eritematoso discoide Comedonic lupus: a rare presentation of discoid lupus erythematosus

Lúpus eritematoso cutâneo crônico é doença autoimune, com apresentações polimorfas que podem, eventualmente, mimetizar outras condições clínicas, causando dificuldade diagnóstica. Quadro acneiforme, com comedões e lesões atróficas pontuadas (pitting scars) pode constituir apresentação atípica do lúpus eritematoso cutâneo discóide<br>Chronic cutaneous lupus erythematosus is a polymorphous autoimmune disease which may mimic some other clinical conditions, causing diagnostic difficulties. Acneiform lesions, including comedones and pitting scars are occasionally atypical presentations of cutaneous discoid lupus erythematosu