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,)(E,\le) and (F,)(F,\le), a monotone map, f:EFf:E\to F is said to be residuated if, and only if, for each yFy\in F, the set {xEf(x)y}\{x\in E\mid f(x)\le y\} has a maximal element, which we denote f #(y)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 ff be residuated interprets as saying it has a left adjoint.

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

ff #id Ff\circ f^\# \le id_F

and

f #fid F.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.)

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

Last revised on September 9, 2021 at 03:54:29. See the history of this page for a list of all contributions to it.