Using the structure of a KZ-monad we create a general categorical workspace in which diagrams can be formally constructed. In particular this abstract framework of category theory is shown to provide a precise semantics for constructing the speciications of complex systems from their component parts.

In fuzzy set theory non-idempotency arises when the conjunction is interpreted by arbitrary t-norms. There are many instances in mathematics where set theory ought to be non-commutative and/or non-idempotent. The purpose of this paper is to combine both ideas and to present a theory of non-commutative and non-idempotent quantale sets (among other things, standard concepts like fuzzy preorders a...

In the eyes of many philosophers, Leibniz established his credentials as a clear and logically precise thinker by having invented the diieren-tial and integral calculus. However, Leibniz's philosophical views about concepts, monads, possible objects, and the containment theory of truth are not often regarded as similar models of clarity and precision. At best, philosophers might agree that Leib...

It was the idea of Calvin Elgot [4] to use Lawvere theories for the study of the semantics of recursion. He introduced iterative theories as those Lawvere theories in which every ideal morphism e : n → n+p (representing a system of recursive equations in n variables and p parameters) has a unique solution, i. e., a unique morphism e† such that the equation e† = [e†, idp] ·e holds. Elgot proved ...

We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg–Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg–Moore monoids coincide with algebras for the list monad transformer (‘done right’) known from Haskell libraries. From this, we obtai...

If S is an order-adjoint monad, that is, a monad on Set that factors through the category of ordered sets with left adjoint maps, then any monad morphism τ : S → T makes T orderadjoint, and the Eilenberg-Moore category of T is monadic over the category of monoids in the Kleisli category of S.

The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad. The goal of this paper is to propos...

For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H. This generalizes the result of Elgot and his coauthors, describing a free iterative theory of a polynomial endofunctor H as the theory R of all rational infinite trees. We present a coalgebraic proof that R i...

We introduce a general construction on 2-monads. develop background maps of 2-monads, their left semi-algebras, and colimits in 2-category. Then, we the colimit induced by map show that obtain structure 2-monad give characterisation its algebras. Finally, apply to 2-monads between free symmetric monoidal cartesian combine them into linear-non-linear 2-monad.

New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursively-defined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a monad.