3
votes
1answer
31 views

Non-monotonic functions on ordered sets

I'm trying to prove that if $~~(A,<_A)~~$ and $~~(B,<_B)~~$ are linearly ordered sets and $~~f: A \rightarrow B~~$ is non-monotonic function than there exist points $~~a,b,c\in A~~$ such that ...
0
votes
1answer
34 views

Could someone check this proof for me?

The problem (and a definition first): The following definition explained Let $U \subseteq \mathbb{N}$. We say that a function $x: \{0,1\}^U\to \{0,1\}$ is a $xor$ on the set $U$ if the following ...
-1
votes
1answer
22 views

partially order

$( A,\le)$ and $(A',\le')$ are partially ordered sets. A map $\phi : A \to A'$ is called order preserving from $(A, \le)$ to $(A', \le')$ if for all $x, y \in A : x \le y \implies \phi(x) \le' ...
0
votes
1answer
57 views

What is the terminology for this type of order-preserving function?

As I roughly know, a function $f$ in $\mathbb{R}$ is called order-preserving if $f(x)>f(y)$ for $x>y$, $x,y \in \mathbb{R}$. May I ask, if anybody knows the terminology for two functions $f$ ...
1
vote
0answers
98 views

Why are complete lattice homomorphisms defined that way?

Given a function $f : X \rightarrow Y$ and a set $A \subseteq X$, there's at least two possible ways of interpreting the direct image $f(A),$ as explained here. In the notation of that question, write ...
4
votes
1answer
132 views

Total orders making a family of functions non-decreasing

Suppose $S$ is a finite set and we are given a family of functions $(f_i:S\to S)_{i\in I}$. When can we find a total order $<$ on $S$ such that all the functions $f_i$ are nondecreasing? I am ...
0
votes
0answers
152 views

From point-wise to essential supremum of a set of real-valued measurable functions

I want to prove some property about essential suprema and I think I can show them for the pointwise supremum $\sup S$. The problem is, that the sets involved are uncountable and thus, the point-wise ...
2
votes
2answers
207 views

Well-Ordered Sets and Functions

I need some assistance with the following problems. I've come up with some ideas but may need some clarification. 1) Let $S$ be a well-ordered set. a) Need to show $S$ has a first element. ...
6
votes
2answers
117 views

Continuation of strictly monotone functions on $\mathbb{R}$

While studying the properties of ordinal utility functions, I came across the following question. Given a strictly increasing function $f : D \rightarrow \mathbb{R}$, where $D$ is an arbitrary ...
5
votes
2answers
75 views

Can such a function exist?

Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$. Does there exist a function $f: \mathbb{R} \to ...
1
vote
0answers
68 views

About function inj, surj and something else. Is this exercise resolved correctly?

This is my problem: For every couple of integers $(a,b)\in\mathbb{Z}\times(\mathbb{Z}\backslash\{0\})$ we denote with $r(a,b)$ the remainder of the division between $a$ and $b$. Consider the ...
2
votes
1answer
143 views

What's the name of this function property?

While playing around with least-fixed-point constructions on a powerset lattice, I've found this property to be useful. Let's say that $F : \mathcal{P(U)} \to \mathcal{P(U)}$, and that $A \subseteq ...