Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.

the paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. motivated by an open question of s. e. rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of l. a. zadeh. on the next step, w...

Given a monad T on Set whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category,...

Simplicial sets, and more generally simplicial objects in a given category, are central to modern mathematics. While I am not a mathematical historian, I thought I would describe in conceptual outline how naturally simplicial sets arise from the classical study of simplicial complexes. I suspect that something like this recapitulates the historical development. We have described simplicial comp...

The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map f : K → L of compact Hausdorff spaces induces a continuous affine map Pf : PK → PL extending P. Together with the canonical embedding ε : K → PK associating to every point its Dirac measure and the barycentric map β associating to every pr...

the notion of absorbent ordered filters in implicative semigroupsis introduced, and its fuzzification is considered. relations among (fuzzy) orderedfilters, (fuzzy) absorbent ordered filters, and (fuzzy) positive implicativeordered filters are stated. the extensionproperty for (fuzzy) absorbent orderedfilters is established. conditions for (fuzzy) ordered filters to be (fuzzy)absorbent ordered ...

Current selection schemes for livestock improvement use a wide variety of phenotypic traits. Some of them, such as sensory, type, or carcass traits, obtain their records from subjective grading performed by trained technicians. Data from this subjective evaluation usually involve classification under a categorical and arbitrary predefined scale, whose output may lead to strong departures from t...

Known upper bounds on the number of required nodes (size) in the ordered binary and multiple-valued decision diagram (DD) for representation of logic functions are reviewed and reduced by a small constant factor. New upper bounds are derived for partial logic functions containing don't cares and also for complete Boolean functions specified by Boolean expressions. The evaluation of upper bounds...

The present paper introduces aggregative spaces and their category AGS, and then establishes a dual adjunction between AGS and the category Agop of aggregation operators on bounded partially ordered sets. Spatial aggregation operators and sober aggregative spaces, enabling us to restrict the dual adjunction between AGS and Agop to a dual equivalence between the full subcategory of Agop consisti...

The classical (Bancilhon-Spyratos) correspondence between view update translations and views with a constant complement reappears more generally as the correspondence between update strategies and meet complements in the order based setting of S. Hegner. We show that these two theories of database view updatability are linked by the notion of “lens” which is an algebra for a monad. We generaliz...