"Monoid objects" without points Let $\mathbf{C}$ be a category with binary Cartesian products. Let's say that a map $f : X \to Y$ is constant if, for every pair of maps $g,h : A \to X$, maps $f \circ g$ and $f \circ h$ are equal. Now, we can define a "monoid object without points" in $\mathbf{C}$ as follows: it is an object $X$ together with a map $* : X \times X \to X$ and a constant map $e : X \to X$ such that $*$ is associative in the usual sense and both compositions $X \xrightarrow{\langle id, e \rangle} X \times X \overset{*}\to X$ and $X \xrightarrow{\langle e, id \rangle} X \times X \overset{*}\to X$ are equal to the identity morphism. We also can define a "group object without points" in a similar way.
If $\mathbf{C}$ has a terminal object and there is a map $x : 1 \to X$, then $X$ is a monoid object since we can define the neutral element as $1 \overset{x}\to X \overset{e}\to X$. Moreover, this defines a bijection between the set of monoid structures and "monoid without points" structures. In particular, the only "monoid objects without points" in $\mathbf{Set}$ are usual monoids and the empty set, but there might be more examples in an arbitrary category.

Is there a name for such "monoid" and "group" objects? Are there any interesting examples of them? Were they considered in the literature?

The reason I'm asking is that (I believe) I have an example of a locale which might not have points but has a structure of a group in the sense I described.
 A: I don't believe there are non-trivial examples of this concept. Assume that $e: X \to X$ is a constant map, then $e$ is idempotent. Any category can be embedded fully faithfully into a Cauchy complete category, so we can assume that $e$ splits as $X \xrightarrow{p} T \xrightarrow{i} X$ with $p\circ i=1_T$. I claim that $T$ is a terminal subterminal object. Indeed, let $f,g: A \to T$, then
$$\begin{eqnarray}
A \xrightarrow f T & = & 
A \xrightarrow f \left( T \xrightarrow i X \xrightarrow p T\right) \xrightarrow 1 \left(T \xrightarrow i X \xrightarrow p T \right) \\
& = &
A \xrightarrow f T \xrightarrow i \left(X \xrightarrow p T \xrightarrow 1 T \xrightarrow i X \right) \xrightarrow p T \\
& = &
A \xrightarrow g T \xrightarrow i \left(X \xrightarrow p T \xrightarrow 1 T \xrightarrow i X \right) \xrightarrow p T \\
& = & A \xrightarrow g T
\end{eqnarray}
$$
Regarding your supposed example of a locale, it is most likely that what you have is either a groupoid in locales or a localic group over some non-trivial base.
