for by the identity morphism and the composition formula in C. We write iff and , i.e. if they are isomorphic in C. Note that in a partial order, any two isomorphic objects are in fact equal. Moving forward, suppose we want to add a monoidal structure to the preorder C. To do so means we must choose an object , called the monoidal unit, and a functor , which we will denote simply by the dot "", called the monoidal multiplication. Thus for any two objects we have an object . We must choose and to be associative and unital, up to isomorphism. This means we must have: and . Furthermore, the fact that · is required to be a functor means—in the present case, where C is a preorder—nothing more than the following: if and then . The additional coherence conditions for monoidal categories are vacuous in this case because every diagram commutes in a preorder. Note that if C is a partial order, the above description is simplified even more, because the associativity and unitality isomorphisms becomes equalities. Another simplification occurs if we assume that the set of objects is the free monoid on a generating set . In this case we could write , where * denotes the Kleene star and the monoidal unit I stands for the empty string. If we start with a set R of generating morphisms (facts about ≤), we recover the usual notion of semi-Thue system, where R is called the "rewriting rule". To return to our example, let N be the category whose objects are the natural numbers 0, 1, 2, ..., with a single morphism if in the usual ordering (and no morphisms from i to j otherwise), and a monoidal structure with the monoidal unit given by 0 and the monoidal multiplication given by the usual addition, . Then N is a monoidal preorder; in fact it is the one freely generated by a single object 1, and a single morphism 0 ≤ 1, where again 0 is the monoidal unit. Properties and associated notions It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using , , , identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that all such diagrams commute. There is a general notion of monoid object