What are the usual definitions of minimality and maximality with respect to an arbitrary relation?

Question

Let $\mathcal R$ be a relation on $S$ and let $T \subseteq S$.

It seems there are two notions floating around of an $\mathcal R$-minimal element of $T$:

1. $x$ is an $\mathcal R$-minimal element of $T$ iff $\forall y \in T: (y \mathrel{\mathcal R} x \implies y = x)$
2. $x$ is an $\mathcal R$-minimal element of $T$ iff $\forall y \in T: y \not\mathrel{\mathcal R} x$

Which of these is more common? Does the other one go by some other name?

The first is compatible with the usual notion of minimality in ordered sets, but the few pages I've found that mention the topic mostly seem to prefer the second.

Answer 1

In my experience (2) is more common when dealing with arbitrary relations. Under this definition a reflexive relation will have no minimal elements, so if you want to use this definition for posets you will want to talk about $<$-minimal elements instead of $\le$-minimal elements. (Here we use the convention that $<$ denotes a strict partial order and $\le$ denotes the corresponding weak partial order.)

People who only deal with posets and prefer to talk about about $\le$ rather than $<$ would probably prefer definition (1). I would call an $x$ as in (1) "weakly minimal" or "minimal" and call an $x$ as in (2) "minimal", "strictly minimal", or "strongly minimal", but I don't know if this agrees with others' usage.