Let $S$ be a semigroup and $I,J$ be two ideals of $S$. For a semilattice we know that $IJ=I\cap J$. Now the question is there a semigroup with the property $IJ=I\cap J$. thanks for your attention



a question about semigroups
I assume you are asking which semigroups have the property that the intersection of any two ideals is their product. 


One condition that will do it is von Neumann regularity. $S$ is regular if, for all $s\in S$, there is $t\in S$ with $sts=s$. 


In any semigroup $IJ\subseteq I\cap J$. If $S$ is regular and $s\in I\cap J$, then choosing $t$ with $sts=s$, we have $s\in I$ and $ts\in J$ and so $s=sts\in IJ$. Thus $IJ=I\cap J$. 


