Perfect semilattices (Q1057890)

We characterize the semilattices S which satisfy the condition (i) \(\forall \theta \in Con(S)\), \(x\theta a_ 1a_ 2\Rightarrow \exists x_ 1,x_ 2\in S:\) \(x_ 1\theta a_ 1\), \(x_ 2\theta a_ 2\) and \(x_ 1x_ 2=x\). Then we characterize the implicative semilattices S which satisfy not only (i) but also the condition (ii) \(\forall \theta \in Con(S)\), \(x\theta a_ 1*a_ 2\Rightarrow \exists x_ 1\), \(x_ 2\in S:\) \(x_ 1\theta a_ 1\), \(x_ 2\theta a_ 2\) and \(x_ 1*x_ 2=x\).