Convex algebras on an interval with semicontinuous monotone operations
This work solves a classification problem in mathematical logic and algebra, but it is incremental as it builds directly on prior results by Mio.
The paper fully classifies convex algebras on the interval [0,1] with monotone and semicontinuous operations, providing an explicit construction of all such operations, which precisely describes the range of theories applicable to Mio's theorem on compact quantitative equational theories.
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.