Let formula_1 be a unital magma, that is, a set with a binary operation formula_2 and an identity element formula_3. If, for formula_4, we have formula_5, then formula_6 is called a left inverse of formula_7 and formula_7 is called a right inverse of formula_6. If an element formula_10 is both a left inverse and a right inverse of formula_11, then formula_10 is called a two-sided inverse, or simply an inverse, of formula_11. An element with a two-sided inverse in formula_1 is called invertible in formula_1. An element with an inverse element only on one side is left invertible or right invertible.