I am using the edition of Bourbaki's "Algebra I" published/printed by Springer in 1989. On p. 11 Bourbaki defines the compatibility between a magma law ⊤ and an equivalence relation R on the magma (E, ⊤) as x ≡ x' and y ≡ y' ⇒ x⊤y ≡ x'⊤y' (all equivalences/congruences are modulo R). So far so good. Let us dub this definition "Def1."
Puzzlingly, after introducing actions (which is just a fancier name for external laws of composition or "external operations"), on pp. 26-27 he defines R to be left compatible with the magma law if it is "compatible with the left action of E on itself derived from" the magma law. To cut a long story short, this comes down to x ≡ y ⇒ z⊤x ≡ z⊤y, for all x, y, z in E (call this "Def2"). Similarly for R right compatible with the magma law: x ≡ y ⇒ x⊤z ≡ y⊤z, for all x, y, z in E (dub this "Def3"). Finally, and this is where the puzzlement kicks in, he says: "For R to be compatible with the law on E [my emphasis] it is necessary and sufficient that it be left and right compatible with the law on E" (call this "Statement1," which is obviously a conjunction of Def2 and Def3).
The big question is, how can Statement1 be interpreted? Is this (a) a definition of compatibility between a magma law and an equivalence relation? It certainly does not look like a definition, but if it is, then what about Def1?? Or is this a (b) theorem, hence Def1 and Statement1 can be shown to be equivalent? If so, I have not managed to find any proofs of this, nor does it look to me to be trivial to derive one from the other. Finally, there may very well be (c) a typo somewhere in Statement1, or somewhere else.
Anyway, please help! Many thanks!
It’s a theorem trivial enough that one might simply call it an observation. If $R$ is left and right compatible with $\top$, $x\equiv x'$, and $y\equiv y'$, then
$$x\top y\equiv x\top y'\equiv x'\top y'\,,$$
so $R$ is compatible with $\top$; the first equivalence follows from left compatibility and the second from right compatibility. The other direction is immediate.