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.
Given posets and , a monotone map, is said to be residuated if, and only if, for each , the set has a maximal element, which we denote .
Each poset gives a small category, and each monotone map gives a functor. From the categorical viewpoint, the condition that be residuated interprets as saying it has a left adjoint.
More exactly, the assignment defines a monotone mapping called the residual mapping. We have
and
The ceiling function is residuated. (The details are given in the discussion at floor.)
One of the standard references for the poset viewpoint is
The Wikipedia entry is
Last revised on September 9, 2021 at 07:54:29. See the history of this page for a list of all contributions to it.