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.
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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 $x*y=x\iff x\le y$.
