example, the associative operad is a symmetric operad generated by a binary operation , subject only to the condition that This condition corresponds to associativity of the binary operation ; writing multiplicatively, the above condition is . This associativity of the operation should not be confused with associativity of composition which holds in any operad; see the axiom of associativity, above. In the associative operad, each is given by the symmetric group , on which acts by right multiplication. The composite permutes its inputs in blocks according to , and within blocks according to the appropriate . The algebras over the associative operad are precisely the semigroups: sets together with a single binary associative operation. The k-linear algebras over the associative operad are precisely the associative k-algebras. Terminal symmetric operad The terminal symmetric operad is the operad which has a single n-ary operation for each n, with each acting trivially. The algebras over this operad are the commutative semigroups; the k-linear algebras are the commutative associative k-algebras. Operads from the braid groups Similarly, there is a non- operad for which each is given by the Artin braid group . Moreover, this non- operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups. Linear algebra In linear algebra, real vector spaces can be considered to be algebras over the operad of all linear combinations . This operad is defined by for , with the obvious action of permuting components, and composition given by the concatentation of the vectors , where . The vector for instance represents the operation of forming a linear combination with coefficients 2,3,-5,0,... This point of view formalizes the notion that linear combinations are the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations. The basic operations of vector addition and scalar multiplication are a generating set for the operad of all linear combinations, while the linear combinations operad canonically encodes all possible operations on a vector space. Similarly, affine combinations, conical combinations, and convex combinations can be considered to correspond to the sub-operads where the terms of the vector sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine