Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint?
Additional discussion. Define a monoid as follows.
Defn 1. A monoid is a triple $(X,*,e)$ such that $*$ is an associative binary operation on $X$, and $e \in X$, and $e$ has the property that for all $x \in X$ it holds that $x * e = e * x = x$.
From here, there's at least two ways of defining a group.
Defn 2. A group is a monoid $(X,*,e)$ such that for all $x \in X$ there exists $y \in X$ such that $x*y=e$.
Defn 2'. A group is a quadruple $(X,*,e,i)$ such that $(X,*,e)$ is a monoid, and $i$ is a function $X \rightarrow X$, and for all $x \in X$ it holds that $x * i(x) = e$.
Now I understand that minimalism favors Defn 2, while practitioners of universal algebra favor Defn 3. However, this is not my question.
My question is not: which is preferable, Defn 2 or Defn 2'?
Rather, my question is: which is preferable, definitions like Defn 2 such that every group is a monoid, or definitions like Defn 2' such that no group is monoid? So just to clarify, I want to know: what is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint?