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.

翻译：在Matteo Mio近期关于紧定量等式理论（此处“紧”指所有推论均可经有限证明推导）的研究中，出现了承载集[0,1]上运算满足单调性及特定半连续性质的凸代数。我们通过给出具备上述性质的[0,1]上所有可能凸运算的显式构造，完整分类此类代数结构。该结果精确刻画了Mio定理适用理论的全体范围。