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Convex algebras on an interval with semicontinuous monotone operations

Ana Sokolova, Harald Woracek

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.

Convex algebras on an interval with semicontinuous monotone operations

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.
Paper Structure (11 sections, 14 theorems, 67 equations)

This paper contains 11 sections, 14 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Monotonicity and continuity properties in convex algebras
  3. Convex algebras
  4. Monotone operations
  5. Semicontinuous operations
  6. Four examples
  7. Structure of convex algebras on $[0,1]$ with (MO), (UC), (LC)
  8. Construction of convex algebras that satisfy (MO), (UC), (LC)
  9. Semicontinuity of homomorphic extensions
  10. Conclusions
  11. Proofs of the main results: Theorems \ref{['E28']}, \ref{['E40']}, \ref{['E15']} and Proposition \ref{['E26']}

Key Result

Lemma 2.8

Let $x_1,\ldots,x_n\in X$ and $p_1,\ldots,p_n\geq 0$ with $\sum_{i=1}^np_i=1$. Let $j\in\{1,\ldots,n\}$ be such that $p_j>0$, and set Then

Theorems & Definitions (35)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 25 more