Let $P$ be a partial order on a finite set $S$ (assume that every element is related to at least one other element besides itself…this raises a few quick questions: is this implied by the definition of partial order and if not, what are the "isolated" points called and what is a partial order with no such points called?) and $R$ be the smallest subset of $(S\times S)\setminus P$ such that for all partial orders $Q\supseteq P$, $$R\cap Q=\varnothing\implies P=Q.$$
Is the relation $R$ unique? Does it have a name? Is it equivalent to something else that does?
I asked a similar (perhaps the same) question in an unwieldy way on MSE at Minimal generating sub-relation of complement of partial order and got no response.
======= edit added by mathematrucker 16 Mar 2022 =======
While searching the literature unsuccessfully for @JosephVanName's nice "one pair extension property of finite partially ordered sets" below it occurred to me that it follows immediately from the fact that the poset under set containment of all unlabeled posets on $n$ points is graded by cardinality. This appears as Lemma 2.1 in New results from an algorithm for counting posets by Culberson and Rawlins (1990), which cites note (3.1) in Aigner, Producing posets (1981). Neither reference supplies a proof, so if anyone knows of one that does, please add it in the comments.
======= edit added by mathematrucker 23 Jul 2022 =======
The following 1988 precursor to Culberson and Rawlins's 1990 paper gives a proof of Lemma 2.1: On counting posets and the structure of the poset of posets. The authors call "isolated" points singletons and posets that have none are said to be in reduced form.