Convex algebras on an interval with semicontinuous monotone operations

TL;DR

This paper classifies convex algebra structures on the interval [0,1] with monotone and semicontinuous operations, clarifying the scope of Matteo Mio's theorem on compact quantitative equational theories.

Contribution

It provides an explicit construction and complete classification of convex operations on [0,1] with the specified properties, extending Mio's theoretical framework.

Findings

Explicit classification of convex operations on [0,1]

Complete description of algebraic structures satisfying the properties

Clarification of the theories covered by Mio's theorem

Abstract

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.