e is the identity element of M. This correspondingly gives a monoid homomorphism. Right monoid actions are defined in a similar way. A monoid M with an action on a set is also called an operator monoid. A semigroup action of S on X can be made into monoid act by adjoining an identity to the semigroup and requiring that it acts as the identity transformation on X. Terminology and notation If S is a semigroup or monoid, then a set X on which S acts as above (on the left, say) is also known as a (left) S-act, S-set, S-action, S-operand, or left act over S. Some authors do not distinguish between semigroup and monoid actions, by regarding the identity axiom () as empty when there is no identity element, or by using the term unitary S-act for an S-act with an identity. The defining property of an act is analogous to the associativity of the semigroup operation, and means that all parentheses can be omitted. It is common practice, especially in computer science, to omit the operations as well so that both the semigroup operation and the action are indicated by juxtaposition. In this way strings of letters from S act on X, as in the expression stx for s, t in S and x in X. It is also quite common to work with right acts rather than left acts. However, every right S-act can be interpreted as a left act over the opposite semigroup, which has the same elements as S, but where multiplication is defined by reversing the factors, , so the two notions are essentially equivalent. Here we primarily adopt the point of view of left acts. Acts and transformations It is often convenient (for instance if there is more than one act under consideration) to use a letter, such as , to denote the function defining the -action and hence write in place of . Then for any in , we denote by the transformation of defined by By the defining property of an -act, satisfies Further, consider a function . It is the same as (see Currying). Because is a bijection, semigroup actions can be defined as functions which satisfy That is, is a semigroup action of on if and only if is a semigroup homomorphism from to the full transformation monoid of . S-homomorphisms Let X and X′ be S-acts. Then