In a recent work of Matteo Mio on compact quantitative equational theories (here compact means that all its consequences are derivable by means of finite proofs) convex algebras on the carrier set [0,1] whose operations are monotone and satisfy certain semicontinuity properties occurred. We fully classify those algebraic structures by giving an explicit construction of all possible convex operations on [0,1] possessing the mentioned properties. Our result thus describes exactly the range of theories to which Mio's theorem applies.
Convex semilattices form an equational class that combines convex algebras and semilattices by a distributive law. For a set $I$ let $\mathcal FI$ be the free convex semilattice with basis $I$. A product $\mathcal FI\times\mathcal FJ$ is generated by $I\times J$, but is not free. Hence, not every map from $I\times J$ into an arbitrary convex semilattice $X$ extends uniquely to a homomorphism from $\mathcal FI\times\mathcal FJ$ to $X$. We investigate a condition on a map $\varphi$ of $I\times J$ into some convex semilattice $X$ that is always necessary and often sufficient for $\varphi$ to have such an extension to a homomorphism $Φ$ of $\mathcal FI\times\mathcal FJ$ into $X$.