The adjunction of a unit to an algebraic structure with a given binary associative operation is discussed by interpreting such structures as semigroups and monoids respectively in a monoidal category. This approach then allows for results on the adjunction of counits to coalgebraic structures with a binary co-associative co-operation as well. Special attention is paid to situations where a give...

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of...

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality betwe...

We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of F! topological spaces. The notions we use generalize those in the adjunction between frames and topological...

In this paper we present a classifier that solves a certain kind of ambiguities in syntactic structure for Spanish, namely, ambiguities as to the point of adjunction of a prepositional phrase in the syntactic structure of a sentence (PP attachment). As a starting point, we used EsTxala dependency grammar for Spanish, integrated within FreeLing, with an accuracy score of 61 % on PP adjunction. O...

For lack of composability of their morphisms, probability spaces, and hence probabilistic automata, fail to form categories; however, they t into the more general framework of precategories, which are introduced and studied here. In particular, the notion of adjunction and weak adjunction for precategories is presented and justiied in detail. As an immediate beneet, a concept of (weak) product ...

For any positive integer $g$, we completely determine the minimal genus function for $\Sigma_{g}\times T^{2}$. We show that lower bound given by adjunction inequality is not sharp some class in $H_{2}(\Sigma_{g}\times T^{2})$. However, construct a suitable embedded surface each and have exact values of functions.

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families Seiberg-Witten equations. This is formulated as a cohomology class on certain abstract simplicial complex consisting embedded surfaces 4-manifold. also give examples which admit positive scalar curvature metrics and for this does not vanish. non-vanishing result our provides new adjunction-type genus constraints confi...