What are some examples of free monoids in physics?

Question

A monoid is given by the following data:

1. A set $S$ of monoid elements,
2. A binary operation, $\cdot$,
3. An identity element $e\in S$, s.t., $\forall a \in S, e \cdot a = a \cdot e = a$
4. Closure: $\forall a,b \in S, \exists c \in S$ s.t. $a \cdot b = c$
5. Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

We can think about free monoid as strings of letters that can be concatenated, though that may not be useful.

I am wondering if someone could give some clear and simple examples of where free monoids appear in physics. My first guess was time series.

Answer 1

An obvious example is the $1$-dimensional Ising model, in which particles in a line exist in one of $2$ spin states, so we'd need a $2$-character alphabet. The Ising model is much more interesting in $2$ or more dimensions, but that would require a multidimensional generalization of your free monoid of strings. (Or you could explore the lattice points in multiple dimensions with a suitable "snaking" motion, so the free monoid is enough.)