Dynamical System Cocycles: (Semi)Group vs. Monoid Definitions - Examples When is Distinction Important?

Let $\mathbb{T}$ be a semigroup ("time") and $X$ a set ("state space") corresponding to a dynamical system (in particular $\mathbb{T}$ acts on $X$ in a known way, $(t , x) \mapsto t \cdot x$). Then given a semigroup $S$, a cocycle is a function $C: \mathbb{T} \times X \to S$ satisfying the following axiom:

$$ \forall t_1, t_2 \in \mathbb{T},\forall x \in X, \quad C(t_2 t_1, x) = C(t_2, t_1 \cdot x) C(t_1, x) \tag{"(Semi)Group Axiom"} \,. $$

When $\mathbb{T}$ and $S$ are also both monoids with identity elements $e_{\mathbb{T}}$, $e_S$ respectively, and the semigroup action of $\mathbb{T}$ on $X$ is also a monoid action i.e. $\forall x \in X$ one has $e_{\mathbb{T}} \cdot x = x$, then one often (e.g. here) sees that the following additional axiom is imposed:

$$\forall x \in X \,, \quad C(e_{\mathbb{T}}, x) = e_S \,. \tag{"Monoid Axiom"} $$

Question: Are there interesting examples of dynamical system cocycles $C: \mathbb{T} \times X \to S$ when $\mathbb{T}$ and $S$ are both monoids but the "Monoid Axiom" is not satisfied? Are there any references discussing these?

I assume probably not but figure I would ask anyway out of curiosity.

Comments/observations: When $\mathbb{T}$ and $S$ are both groups and the semigroup action of $\mathbb{T}$ is also a group/monoid action (i.e. we still have for all $x \in X$ that $e_{\mathbb{T}} \cdot x = x$) then the "(Semi)Group Axiom" implies the "Monoid Axiom" for reasons completely analogous to why semigroup homomorphisms are monoid homomorphisms when the monoids in question are also groups:

$$C(e_{\mathbb{T}}, x) = C(e_{\mathbb{T}}^2, x) = C(e_{\mathbb{T}}, e_{\mathbb{T}} \cdot x) C(e_{\mathbb{T}} , x) = C(e_{\mathbb{T}}, x)^2 \,,$$

so $C(e_{\mathbb{T}}, x)$ is idempotent in $S$, and because $S$ is a group that means $C(e_{\mathbb{T}}, x ) = e_S$. This appears to be why Terence Tao only gives the "(Semi)Group Axiom" in his post about cocycles of dynamical systems, because there $\mathbb{T}$ and $S$ are both assumed to be (abelian) groups.

The motivation for this question comes from the definition of "random dynamical system" (Definition 1.1.1) of Ludwig Arnold's Random Dynamical Systems. There the "(Semi)Group Axiom" is equation (1.1.2), and the "Monoid Axiom" is equation (1.1.1). The latter axiom is only required when $\mathbb{T}$ is a monoid (in that context $S$ is always a monoid), and the same observation is made about the "Monoid Axiom" being redundant when we can assume $\mathbb{T}$ and $S$ are both groups.

Also note that the "(Semi)Group Axiom" does not say that $C$ defines an $X$-indexed family of semigroup homomorphisms. That would only be true if we had $C(t_2 t_1, x) = C(t_2, x) C(t_1, x)$, i.e. if $t \cdot x = x$ for all $t \in \mathbb{T}$, $x \in X$, i.e. if the semigroup action of $\mathbb{T}$ on $X$ were trivial. (That is what is meant in this answer when it says "Note that a cocycle independent of $x$ is given by a homomorphism $G \to H$", because there $\mathbb{T} = G$ and $S=H$.) But of course the axiom is very similar, which is why I am calling it the "(Semi)Group Axiom".

Related question: Intuition of cocycles and there use in dynamical systems

Due to abstract nonsense, I would say for the purposes of dynamics any semigroup action of a monoid can be considered to be a monoid action (indeed any semigroup action of a semigroup can be considered to be a monoid action of a monoid, by adding an extra element if necessary to be and act as identity).

Say $(M,e)$ is a semigroup with identity (i.e. monoid), $X$ is a set, $\alpha_\bullet:M\to \operatorname{End}_{\text{Set}}(X)$ is a family of self-maps of $X$ satisfying the semigroup property

$$\alpha_{ts}=\alpha_t\circ \alpha_s.$$

Like you said it is not necessarily the case that $\alpha_{e}=\operatorname{id}_X$. However the semigroup property still limits the possibilities for $\alpha_e$, in that:

$$\alpha_e\in\{f\in\operatorname{End}_{\text{Set}}(X)\,|\, \forall t\in M: \alpha_t\circ \alpha_e=\alpha_t=\alpha_e\circ \alpha_t\},$$

in particular $\alpha_e$ has to be idempotent:

$$\alpha_e\circ \alpha_e=\alpha_e.$$

For $f:X\to X$ an idempotent, define an equivalence relation $\sim_f$ on $X$:

$$x\sim_f y\iff x=y \quad\text{ or }\quad f(x)=y\quad\text{ or }\quad x=f(y) \quad\text{ or }\quad f(x)=f(y).$$

(Reflexivity and symmetry of this relation is straightforward; for transitivity one can draw a $4\times 4$ table whose columns are all possibilities for $x\sim_f y$ and whose rows are all possibilities for $y\sim_f z$; idempotence of $f$ is important here. Succinctly, $y\sim_f x \iff y\in f^{-1}(f(x))$, that is, $y$ is in the $f$-saturation of $x$.)

Let us denote by $\pi^f:X\twoheadrightarrow X/\sim_f$ the canonical projection to the associated quotient set. Since $f(x)=f(x)$, $f(x)\sim_f x$, so that $f/\sim_f = \operatorname{id}$, i.e.

In particular, for $f=\alpha_e$, $\underline{\alpha_e}: X/\sim_{\alpha_e}\to X/\sim_{\alpha_e}, [x]\mapsto [\alpha_e(x)]$ is the identity map. Finally for $t\in M$, $\alpha_t$ preserves $\sim_{\alpha_e}$, so that

$$\underline{\alpha_t}: X/\sim_{\alpha_e}\to X/\sim_{\alpha_e},\,\, [x]\mapsto [\alpha_t(x)]$$

is well-defined. It is straightforward that now we have a monoid action $\underline{\alpha_\bullet}: (M,e)\to \left(\operatorname{End}_{\text{Set}}(X/\sim_{\alpha_e}),\operatorname{id}_{X/\sim_{\alpha_e}}\right)$.

My estimation is that (possibly with some natural modifications) this argument can be extended to actions with structure (measurable, topological, smooth, ...). So it is benign to assume that any semigroup action of a monoid is in fact a monoid action.

Finally note that in dynamics literature monoid actions are at times extended without notice to group actions even (using some version of the natural extension/ inverse limit/ solenoid construction discussed at Proving a.e surjectivity of suggested factor map in Natural extension of Standard Borel dynamical system, Does any measure preserving system have an invertible extension?, Projective limit of spaces of probability measures is bijective to the space of probability measures on a projective limit., or the aforementioned book of Arnold, pp.547-548).