An order-theoretic interpretation of Jacobsthal multiplication that makes associativity obvious

So, here's something that had bugged me for a long time. Jacobsthal multiplication is associative, but the proof of this is unsatisfying. Normally the way you prove an operation is associative is that you show that a multiary version of it makes sense; you don't just literally show that a(bc)=(ab)c, you show that abc makes sense on its own as a ternary thing (or with even more operands) and that…