square. Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table the only element with a two-sided inverse is the identity element 1. If the operation is associative then if an element has both a left inverse and a right inverse, they are equal. In other words, in a monoid (an associative unital magma) every element has at most one inverse (as defined in this section). In a monoid, the set of invertible elements is a group, called the group of units of , and denoted by or H1. In a semigroup The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity; that is, in a semigroup. 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. In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given in this section. Only elements in the Green class H1 have an inverse from the unital magma perspective, whereas for any idempotent e, the elements