nLab generalized inverse

Idea

In algebra, matrix theory, operator theory, approximation theory and some other areas where a multiplicative binary operation is defined a two-sided inverse of some elements is often either non-existent or ill-behaved, hence some generalized notion may be often useful. Generalized inverse would still retain some of the properties.

Examples

Inverses in inverse semigroups

In inverse semigroups, every element AA has a generalized inverse A 1A^{-1}, called just an inverse in that setup, with defining properties

AA 1A=A A A^{-1} A = A
A 1AA 1=A 1 A^{-1} A A^{-1} = A^{-1}

Moore-Penrose inverse?

Using approximate unit

Some nonunital operator algebras have an approximate unit; inverses can be defined using an approximate unit instead of the usual unit.

Literature

Created on May 23, 2024 at 09:30:50. See the history of this page for a list of all contributions to it.