and thus that the expression is unambiguous. Identity axiom The identity axiom (for a binary operation) can be visualized in a tree as: meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference. As for categories, is a corollary of the identity axiom. Examples Endomorphism operad in sets and operad algebras The most basic operads are the ones given in the section on "Intuition", above. For any set , we obtain the endomorphism operad consisting of all functions . These operads are important because they serve to define operad algebras. If is an operad, an operad algebra over is given by a set and an operad morphism . Intuitively, such a morphism turns each "abstract" operation of into a "concrete" -ary operation on the set . An operad algebra over thus consists of a set together with concrete operations on that follow the rules abstractely specified by the operad . Endomorphism operad in vector spaces and operad algebras If k is a field, we can consider the category of finite-dimensional vector spaces over k; this becomes a monoidal category using the ordinary tensor product over k. We can then define endomorphism operads in this category, as follows. Let V be a finite-dimensional vector space The endomorphism operad of V consists of = the space of linear maps , (composition) given , , ..., , their composition is given by the map , (identity) The identity element in is the identity map , (symmetric group action) operates on by permuting the components of the tensors in . If is an operad, a k-linear operad algebra over is given by a finite-dimensional vector space V over k and an operad morphism ; this amounts to specifying concrete multilinear operations on V that behave like the operations of . (Notice the analogy between operads&operad algebras and rings&modules: a module over a ring R is given by an abelian group M together with a ring homomorphism .) Depending on applications, variations of the above are possible: for example, in algebraic topology, instead of vector spaces and tensor products between them, one uses (reasonable) topological spaces and cartesian products between them. "Little something" operads The little 2-disks operad is a topological operad where consists of ordered lists of n disjoint disks inside the unit disk of centered at the origin. The symmetric group acts on such configurations