In a semigroup "S" an element "x" is called (von Neumann) regular if there exists some element "z" in "S" such that "xzx" = "x"; "z" is sometimes called a pseudoinverse. An element "y" is called (simply) an inverse of "x" if "xyx" = "x" and "y" = "yxy". Every regular element has at least one inverse: if "x" = "xzx" then it is easy to verify that "y" = "zxz" is an inverse of "x" as defined in this section. Another easy to prove fact: if "y" is an inverse of "x" then "e" = "xy" and "f" = "yx" are idempotents, that is "ee" = "e" and "ff" = "f". Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and "ex" = "xf" = "x", "ye" = "fy" = "y", and "e" acts as a left identity on "x", while "f" acts a right identity, and the left/right roles are reversed for "y". This simple observation can be generalized using Green's relations: every idempotent "e" in an arbitrary semigroup is a left identity for "Re" and right identity for "Le". An intuitive description of this fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.