Monoid

A term used as an abbreviation for the phrase "semi-group with identity" . Thus, a monoid is a set $ M $ with an associative binary operation, usually called multiplication, in which there is an element $ e $ such that $ ex = x = xe $ for any $ x \in M $. The element $ e $ is called the identity (or unit) and is usually denoted by $ 1 $. In any monoid there is exactly one identity. If the operation given on the monoid is commutative, it is often called addition and the identity is called the zero and is denoted by $ 0 $.

Examples of monoids.

1) The set of all mappings of an arbitrary set $ S $ into itself is a monoid relative to the operation of successive application (composition) of mappings. The identity mapping is the identity.

2) The set of endomorphisms of a universal algebra $ A $ is a monoid relative to composition; the identity is the identity endomorphism.

3) Every group is a monoid.

Every semi-group $ P $ without an identity can be imbedded in a monoid. For this it suffices to take a symbol $ 1 $ not in $ P $ and give a multiplication on the set $ P \cup \{ 1 \} $ as follows: $ 1 \cdot 1 = 1 $, $ 1 \cdot x = x = x \cdot 1 $ for any $ x \in P $, and on elements from $ P $ the operation is as before. Every monoid can be represented as the monoid of all endomorphisms of some universal algebra.

An arbitrary monoid can also be considered as a category with one object. This allows one to associate with a monoid $ M $ its dual (opposite, adjoint) monoid $ M ^ { \mathop{\rm op} } $. The elements of both monoids coincide, but the product of $ x $ and $ y $ in $ M ^ { \mathop{\rm op} } $ is put equal to the product $ yx $ in $ M $.

The development of the theory of monoids and adjoint functors has shown the utility of the definition of a monoid in so-called monoidal categories. Suppose given a category $ \mathfrak M $ equipped with a bifunctor $ \otimes : \mathfrak M \times \mathfrak M \rightarrow \mathfrak M $, an object $ Z $ and natural isomorphisms

$$ \alpha _ {ABC} : ( A \otimes B ) \otimes C \rightarrow A \otimes ( B \otimes C ) , $$

$$ \lambda _ {A} : Z \otimes A \rightarrow A ,\ \rho _ {A} : A \otimes Z \rightarrow A , $$

satisfying coherence conditions. An object $ M $ is called a monoid in the category $ \mathfrak M $ if there are morphisms $ \mu : M \otimes M \rightarrow M $ and $ \epsilon : Z \rightarrow M $ such that the following diagrams are commutative:

$$ $$

If $ \mathfrak M $ is taken to be the category of sets, $ \otimes $ the Cartesian product, $ Z $ a one-point set, and the isomorphisms $ \alpha $, $ \lambda $ and $ \rho $ are chosen in the natural way ( $ \alpha (( a , b ) , c )= ( a , ( b , c )) $, $ \lambda ( z , a ) = a = \rho ( a , z ) $), then the second definition of a monoid turns out to be equivalent to the original definition.

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2]	S. MacLane, "Categories for the working mathematician" , Springer (1971)

Comments

For monoidal categories, and particularly the coherence conditions that the isomorphisms $ \alpha _ {ABC } $, $ \lambda _ {A} $ must satisfy cf. [1], Chapt. 7, Sects. 1-2.