Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, is a (partial) order.

Question. Does the order $\le$ have a standard name? Is there any hope to trace the first publication where $\le$ was explicitly defined?

The restriction of $\le$ to the idempotents of $S$ plays an important role in the structure theory of semigroups (and rings). In this regard, one of the earliest references I know of is [D. McLean, Idempotent Semigroups, Amer. Math. Monthly 61 (1954), No. 2, 110–113]. However, there is no explicit definition of the order $\le$ in McLean's work.

On the other hand, the order $\le$ is explicitly considered by G. Birkhoff in Lemma I.6 of the 3rd edition of his classical monograph on lattice theory (AMS, 1973). Yet, Birkhoff's definition is restricted to semilattices (that is, idempotent commutative semigroups). I don't know if the lemma is already present in the 1st or 2nd editions of the book.