This work presents a detailed analysis of the combinatorics modular operads. These are operad-like structures that admit contraction operation as well an operadic multiplication. Their governed by graphs cycles, and known for their complexity. In 2011, Joyal Kock introduced powerful graphical formalism paper extends work. A monad operads is constructed corresponding nerve theorem proved, using ...

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...

In this work we generalize previous constructions of fuzzy set categories, introduced in [5], by considering L-fuzzy sets in which the values of the characteristic functions run on a completely distributive lattice, rather than in the unit real interval. Later, these L-fuzzy sets are used to define the L-fuzzy categories, which are proven to be rational. In the final part of the paper, the L-fu...

We present a (co)algebraic treatment of iteration-free dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL), both without star. The main observation is that the program/game constructs of PDL/GL arise from monad structure, and the axioms of these logics correspond to certain compatibilty requirements between the modalities and this monad structure. Our main contrib...

Given an arbitrary locally finitely presentable category K and finitary monads T and S on K , we characterize monad morphisms α : S −→ T with the property that the induced functor α∗ : K T −→ K S between the categories of Eilenberg-Moore algebras is fully faithful. We call such monad morphisms dense and give a characterization of them in the spirit of Beth’s definability theorem: α is a dense m...

We present an interface from Haskell to quantum programming: the Quantum IO monad, and use it to implement Shor’s factorisation algorithm. The QIO monad separates reversible (i.e. unitary) and irreversible (i.e. probabilistic) computations and provides a reversible let operation (ulet), allowing us to use ancillas (auxiliary qubits) in a modular fashion. Exploiting Haskell’s class system we can...