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 $A$ has a generalized inverse $A^{-1}$, called just an inverse in that setup, with defining properties

$$A A^{-1} A = A$$

$$A^{-1} A A^{-1} = A^{-1}$$

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

• wikipedia: generalized inverse, Moore–Penrose inverse
• C. R. Rao, S. K. Mitra, Generalized inverse of matrices and its applications, Wiley 1971.