## 1. Introduction

## 2. Preliminaries

## 3. Results

#### 3.1. Threshold Representation

**Proposition**

**1.**

- 1.
- $<{\succ}_{S},\succ >$ is a nested system of strict partial order.
- 2.
- $<{\succ}_{S},\succ >$ admits a general threshold representation.

**Proof.**

**Corollary**

**1.**

- 1.
- c satisfies Weak Expansion AND Aizerman Expansion, and its associated base relation is acyclic.
- 2.
- There exist a utility function $u:X\to \mathcal{R}$ and a threshold function $\theta :\Phi \to {\mathcal{R}}^{+}$ satisfying (1), (2), and (3) such that$$c\left(S\right)=\{y\in S:/\phantom{\rule{-4.33333pt}{0ex}}\exists x\in S,\phantom{\rule{4pt}{0ex}}s.t.\phantom{\rule{4pt}{0ex}}u(x)-u(y)>\theta (x,y,S\left)\right\}$$

**Example**

**1.**

#### 3.2. Multi-Dimensional Utility and Aggregation

Frequently, a course of action satisfying a number of constraints, even a sizeable number, is far easier to discover than a course of action maximizing some function.

**Proposition**

**2.**

- 1.
- $<{\succ}_{S},\succ >$ is a nested system of strict partial order.
- 2.
- $<{\succ}_{S},\succ >$ admits a multi-utility aggregation representation.

**Proof.**

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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