nLab residuated morphism

Idea

Coming originally from the theory of (partially) ordered sets, the notion of residuated morphism or residuated mapping is, from a categorical viewpoint, just the condition that a map of posets when considered as a functor between the corresponding categories, has a left adjoint.

The notion is used in the theory of idempotent semirings to give a form of `pseudo-solution' to equations which fail to have actual solutions.

Definition

Given posets $(E,\le)$ and $(F,\le)$, a monotone map, $f:E\to F$ is said to be residuated if, and only if, for each $y\in F$, the set $\{x\in E\mid f(x)\le y\}$ has a maximal element, which we denote $f^\#(y)$.

Translation into categorical terms

Each poset gives a small category, and each monotone map gives a functor. From the categorical viewpoint, the condition that $f$ be residuated interprets as saying it has a left adjoint.

More exactly, the assignment $y\mapsto f^#(y)$ defines a monotone mapping $f^\#:F\to E$ called the residual mapping. We have

$$f\circ f^\# \le id_F$$

and

$$f^\#\circ f \ge id_F.$$

Standard example

The ceiling function $\lceil{-}\rceil:\mathbb{R}\to \mathbb{Z}$ is residuated. (The details are given in the discussion at floor.)

Related entries

• residuated lattice

• residuated idempotent semiring

• Galois connection

References.

One of the standard references for the poset viewpoint is

• T. S. Blyth and M. F. Janowitz, Residuation Theory, Pergamon Press, 1972.

The Wikipedia entry is

• residuated mapping