Convex algebras on an interval with semicontinuous monotone operations
Ana Sokolova, Harald Woracek

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.
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Taxonomy
TopicsFunctional Equations Stability Results · Optimization and Variational Analysis · Mathematical and Theoretical Analysis
