This exercise asks me to verify the axiom of a really weirdly defined category:
Define a category $\mathcal{C}$ as follows:
$(1)$ There is only one object $*$;
$(2)$ The set of morphism $Mor(*,*)$ consists of the algebraic terms $t:=x_{0}|f(t)|g(t,t)$, where $x_{0}$ is a given variable and $f, g$ are two given function symbols;
$(3)$ The composition law is a substitution recursively defined as $t(t')$, where $t'$ replace $x_{0}$ in $t$.
The exercise then asks me to check the axioms and to describe the set of all isomorphism $Iso(*,*)$.
Well the first axiom is trivial true since there is only object.
Then we just need to check that the existence of two-sided identity element in $Mor(*,*)$ and check the associativity of the composition.
Since the construction of composition is really weird, I tried to compute one composition and to see what is going on. However, after the computation, I had no idea if this composition is even well-defined
For $t_{1}, t_{2}\in Mor(*,*)$, by definition we have $$t_{1}:=x_{0}|f(t_{1})|g(t_{1}, t_{1})\ \text{and}\ t_{2}:=x_{0}|f(t_{2})|g(t_{2}, t_{2}),$$ so that by the construction of the law of composition, we have \begin{align*} t_{2}\circ t_{1}:=t_{2}(t_{1})&=t_{1}|f(t_{2})|g(t_{2}, t_{2})\\ &=x_{0}|f(t_{1})|g(t_{1}, t_{1})|f(t_{2})|g(t_{2}, t_{2})\\ &=x_{0}|f(t_{2})f(t_{1})|g(t_{2}, t_{2})g(t_{1}, t_{1}). \end{align*}
However, I still need to show $|f(t_{2})f(t_{1})|=|f(t_{2}\circ t_{1})|$ and $g(t_{2}, t_{2})g(t_{1}, t_{1})=g(t_{2}\circ t_{1}, t_{2}\circ t_{1})$, right? Because if the composition is well-defined, we should have $$t_{2}\circ t_{1}=x_{0}|f(t_{2}\circ t_{1})|g(t_{2}\circ t_{1}, t_{2}\circ t_{1})\in Mor(*,*),$$ right?
But I don't see any way to show this. Also, what is the identity element in this case? Thank you!
As is well-known, a category with just one object is actually a monoid, no more or less. I think that the formal definition of morphism as $\, t:=x_{0}|f(t)|g(t,t) \,$ seems like a simplified form of BNF definition of a simple grammar. What it essentially states is that the language consists of a single variable denoted by $\,x_0,\,$ and one (or more) unary functions denoted by $\,f\,$ and one (or more) binary functions denoted by $\,g.\,$ An example of such a language expressions is $\,g(x_0,f(x_0))\,$ which denotes the function defined by $\, x_0 \mapsto g(x_0,f(x_0)).\,$ The composition law of the category (monoid) is just composition of the functions which the expressions denote, and of course, function composition is always associative.