# If a binary operation induces a order-semilattice, is it an algebraic-semilattice?

If a magma $$(S;*)$$ satisfies associativity, commutativity, and idempotence, then the relation $$xRy$$ defined by $$x * y = x$$ is a meet-semilattice. I want to know if the converse is true. That is, if, in a magma $$(S;*)$$, the binary relation defined by $$x*y=x$$ is a meet-semilattice, then is $$*$$ associative, commutative, and idempotent? Well, it has to be idempotent, but does it have to also be associative and commutative? I would be very interested in a counterexample that is neither commutative nor associative, should one exist.

Let $$(S,\le)$$ be a $$3$$-element chain with elements $$a\lt b\lt c$$. Define $$x*y=x$$ if $$x\le y$$ and $$x*y\in S\setminus\{x,y\}$$ if $$x\gt y$$. The magma $$(S,*)$$ is noncommutative since $$a*b=a\ne c=b*a$$, nonassociative since $$(b*b)*a=b*a=c\ne b=b*c=b*(b*a)$$.
More generally, let $$(S,\le)$$ be almost any meet-semilattice, and let $$*$$ be almost any binary operation on $$S$$ such that $$x*y=x\iff x\le y$$.