20260319 - The monoids. (#1 of a series about Universal Algebra)

"They are useless," said a professor of mine once. Some people certainly consider them to be so. Many find them incomplete and feel that they’re lacking something important. They’re crippled groups. They’re monoids.

Think of any set of objects. Call it $M$. It doesn’t matter. Operations can be defined in it as mappings like $f : M^n \rightarrow M$, where $n$ is a natural number, and $M^n$ is the set of all n-tuples of elements of A. It’s also defined that if $n = 0$, then $M^0 = \{\varnothing\}$. These mappings are called n-ary operations, and all monoids have both a binary and a nullary operation. Also called constants, nullary operations just fixate an element of $M$.

Two operations may seem wrong for some, but here we’ll treat monoids as universal algebras. That is, a pair $(M, F)$ of a non-empty set $M$, the $universe$ of the algebra, and a family of operations $F$ on $M$, indexed by some set $I$. The fundamental or basic operations $f_i \in F$ in our case are a binary operation that takes $a,b \in M$ and associates them to $a \cdot b$ and a nullary operation that picks out $e \in M$.

Three identities are also needed. These being the associative law $(x \cdot y)\cdot z = x \cdot (y \cdot z)$ and two identities for the neutral element $e$, which are $e \cdot x = x$ and $x \cdot e = x$, satisfied by any $x,y,z \in M$. Treat identities as axioms if you like, but bear in mind that they are equations satisfied for any "substitution" of elements of $M$ in place of the "variables" $x,y$ and $z$. To not get too ahead of myself, I won’t show the formal reasoning behind identities yet, only a notation. We can denote $M \models \{j_1, j_2,j_3\}$, where $j_1,j_2$ and $j_3$ are the identities I just described, to say that M satisfies those identities.

The professor who considered monoids useless also stated recently that "algebra is a simple science." Considering the definitions given so far, we can easily consider what a subalgebra of an algebra is, or more specifically, what a submonoid of a monoid is. We say that $N \subseteq M$ is a subalgebra of $M$ if $N$ is an algebra with the operations defined in $M$, or in other words, if $N$ is closed under the operations defined in $M$. That is, if $M$ is a monoid, we consider $N \subseteq M$ a submonoid when $e \in N$, and for all $a,b \in N$, $a \cdot b \in N$. Note that $N \models \{j_1, j_2, j_3\}$ is immediate since $N$ is a subset of $M$.

To say that monoids are crippled groups might be a little mean, but examples of monoids usually arise from structures where the underlying set is just "inverses short" of being a group. The natural set $\mathbb{N}$ with the operation of sum and the set $M_n(K)$ of all matrices of order $n$ in a field $K$ with the operation of product of matrices both fail to be groups by a tweak in their underlying sets. There are many other examples like that.

They probably aren’t completely useless, I think. Perhaps a bit too fundamental, too simple for you to care, but every so often I think of them.