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.
Proofs in quantitative algebra are, generally, infinite objects due to the presence of infinitary rules in the deductive apparatus. A quantitative equational theory is called compact if all its consequences are derivable by means of finite proofs, that is, proofs not using any instances of the infinitary rules. This definition is purely logical. In this work we give an equivalent, but categorical, characterisation based on the notion of monad restriction.