On Double Dualization Monads.

Monads in double categories

Introduction The development of the formal theory of monads, begun in [23] and continued in [15], shows that much of the theory of monads [1] can be generalized from the setting of the 2-category Cat of small categories, functors and natural transformations to that of a general 2-category. The generalization, which involves defining the 2-category Mnd(K) of monads, monad maps and monad 2-cells ...

Non-monotone Dualization via Monotone Dualization

The non-monotone dualization (NMD) is one of the most essential tasks required for finding hypotheses in various ILP settings, like learning from entailment [1, 2] and learning from interpretations [3]. Its task is to generate an irredundant prime CNF formula ψ of the dual f where f is a general Boolean function represented by CNF [4]. The DNF formula φ of f is easily obtained by De Morgan’s la...

Monads on Projective Space

Monads on Composition Graphs

Collections of objects and morphisms that fail to form categories inas-much as the expected composites of two morphisms need not always be deened have been introduced in 14, 15] under the name composition graphs. In 14, 16], notions of adjunction and weak adjunction for composition graphs have been proposed. Building on these deenitions, we now introduce a concept of monads for composition grap...

Monads on Dagger Categories

The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleis...