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Partial representations of orderings

Gianni Bosi, Asier Estevan, Magali Zuanon

TL;DR

A new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular) and the advantages of this kind of representations are presented, particularly in the case of partial orders and semiorders.

Abstract

In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability,...) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.

Partial representations of orderings

TL;DR

A new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular) and the advantages of this kind of representations are presented, particularly in the case of partial orders and semiorders.

Abstract

In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability,...) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.
Paper Structure (6 sections, 8 theorems, 18 equations, 8 figures)

This paper contains 6 sections, 8 theorems, 18 equations, 8 figures.

Table of Contents

  1. Introduction and motivation
  2. Notation and preliminaries
  3. Partial representability of preorders
  4. Partial representability of intransitive relations
  5. Partial Scott-Suppes representations
  6. Further comments

Key Result

Proposition 17

Let $\precsim$ be a preorder on $X$ and $A$ the set of all isolated points. Assume that there exists a (continuous) partial multi-utility representation $\{v_i\}_{i\in I}$. If there exists a (continuous) Richter-Peleg utility of the preorder on a subset ${Y}$ such that $X\setminus A \subseteq Y$, th

Figures (8)

  • Figure 1: Partially ordered set with its corresponding labelings.
  • Figure 2: Partially ordered set with a partial Richter-Peleg multi-utility.
  • Figure 3: The random structure and a partial Richter-Peleg multi-utility representation of a poset.
  • Figure 4: A continuous partial Richter-Peleg multi-utility representation of a poset.
  • Figure 5: Three partial orders and the corresponding Scott topologies.
  • ...and 3 more figures

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Remark 9
  • Definition 10
  • ...and 42 more