The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a $c$-colouring of that promised to be $k$-colourable, where $c\geq k$. This problem naturally generalises promise homomorphism problems and further constraint satisfaction problems. these has recently been studied through an algebraic approach. In this paper, we introduce two new techniques ...

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...

Conceiving of premises as collected into sets or multisets, instead of sequences, may lead to triviality for classical and intuitionistic logic in general proof theory, where we investigate identity of deductions. Any two deductions with the same premises and the same conclusions become equal. In terms of categorial proof theory, this is a consequence of a simple fact concerning adjunction with...

In this paper, we consider some key characteristics that AI should exhibit to enable hybrid agencies include subject-matter experts and their AI-enabled decision aids. We will hint at the design requirements of guaranteeing tools are: open, multiple, continuous, cautious, vague, analogical and, most importantly, adjunct with respect decision-making practices. argue especially adjunction is an i...

in this work, we describe an adjunction between the comma category of set-based monads under the v -powerset monad and the category of associative lax extensions of set-based monads to the category of v -relations. in the process, we give a general construction of the kleisli extension of a monad to the category of v-relations.

The polar curves of foliations F having a curve C of separatrices generalize the classical polar curves associated to hamiltonian foliations of C. As in the classical theory, the equisingularity type ℘(F) of a generic polar curve depends on the analytical type of F , and hence of C. In this paper we find the equisingularity types ǫ(C) of C, that we call kind singularities, such that ℘(F) is com...

It’s widely assumed—often tacitly—that A-movement doesn’t move through intermediate positions where it doesn’t seem to check features (e.g. Baltin 2001). Chomsky (2001), however, proposes that all movement from inside the complement of v to a position above vP must pass through an intermediate position in the edge of vP. Within Chomsky’s proposal, this follows from the assumption that vP is a P...

The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional abstraction and application to an argument in the postulates of the lambda calculus. Such an inversion principle arises also in two adjoint situations involving ...