One common definition for minimal element is as follows (I am aware of equivalent definitions):
Let $\mathcal{R}$ be a partial order relation on a set $S$, and let $S_0\!\subseteq\!S$. An element $m\!\in\!S_0$ is a minimal element of $S_0$ when: $$\forall s\!\in\!S_0\big[\ s\mathcal{R}m\rightarrow s\!=\!m\ \big].$$
I understand the intuitive meaning of this concept in the context of a given partial order as well as how it is used to define wellfoundedness. But does this definition have meaning in a more general context for an arbitrary relation (i.e., for a relation that is not necessarily a partial order)?
In other words, can a minimal element be defined as:
Let $\mathcal{R}$ be any relation on a set $S$, and let $S_0\!\subseteq\!S$. An element $m\!\in\!S_0$ is a minimal element of $S_0$ when: $$\forall s\!\in\!S_0\big[\ s\mathcal{R}m\rightarrow s\!=\!m\ \big].$$
Yes. You can define this for any relation. It's just that when considering order relations we have a geometric intuition, and so the term "minimal" makes sense.
For an arbitrary relation, especially non-transitive ones, it might come across as somewhat awkward to try and understand how exactly an element satisfying this property is minimal. But once you forgo the lack of intuition, you realize that it is just a definition.