 Conversely, if "S" is defined by (1), then "xax" is an inverse for "a", since "a"("xax")"a" = "axa"("xa") = "axa" = "a" and ("xax")"a"("xax") = "x"("axa")("xax") = "xa"("xax") = "x"("axa")"x" = "xax".The set of inverses (in the above sense) of an element "a" in an arbitrary semigroup "S" is denoted by "V"("a").Thus, another way of expressing definition (2) above is to say that in a regular semigroup, "V"("a") is nonempty, for every "a" in "S".