All Questions
Tagged with lattice-orders fixed-point-theorems
24
questions
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34
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Prove equivalence between properties of relations using fixpoint calculus
The problem
Let $a$ be a binary endorelation of some countable set $S$, i.e. $a \subseteq S \times S$ .
I need to show that the following properties are equivalent:
$ (a^*)^{-1} ; a^* \subseteq a^* ; ...
- 550
1
vote
1
answer
79
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Fixed point problem involving a minimization
Let $X \subset {\Bbb R}^n$ be a compact and convex set, and let $f : X \times X \to {\Bbb R}$ be a continuous and differentiable scalar field defined by
$$
f(x,x^*)=g(x) + \sum_{i=1}^n x_ih_i(x^*).
$$
...
- 1,400
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1
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33
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Least fixed point and monotone mappings in complete lattice
Let $(E_1,\leq_1)$ and $(E_2,\leq_2)$ be two complete lattices. also let $f_1 : E_1 \times E_2 \rightarrow E_1$ and $f_2 : E_1 \times E_2 \rightarrow E_2$ be mappings monotonic with respect to their ...
- 33
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60
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Generalized Tarski fixed point theorem to multivariate functions
I wish to know if a theorem like the following exists, or if there are any obvious counterexamples. This theorem generalizes Tarski's fixed point theorem in the case of powersets.
Let $X,Y$ be non-...
- 175
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59
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Understanding Tarski's fixed-point theorem.
I changed my question slightly.
(Tarski Fixed Point Theorem). Let $X=\prod^{N}_{i=1} X_{i}$ where each $X_{i}$ is a compact interval of $\mathbb{R}$. Suppose $\phi : X \rightarrow X$ is an increasing ...
0
votes
3
answers
318
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Proof of general Knaster-Tarski fixpoint theorem
Prove that the set of fixed points $Fix(f)$ of an order-preserving operator $f$ on a complete lattice
$(L, \sqsubseteq)$ is a complete lattice itself. Moreover, show that $\mathrm{Fix}(f)$ is a ...
- 101
1
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1
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162
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Tarski Fixed Point Theorem Counterexample?
I have a quick question regarding Tarski's fixed point theorem. It states that for all order-preserving $f:X\to X$, the set of fixed points must be a complete lattice. I was wondering how that would ...
- 607
1
vote
1
answer
243
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Fixed points in a complete lattice
Let $(X,\leq)$ be a complete lattice, $\perp,\top$ be the least and the greatest element of $X$ and $f:X\to X$ is a monotonic function.
Tarski's fixed point theorem states that the set of fixed ...
- 427
1
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0
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24
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Is this simple demand-based prices game a submodular game?
I have this simple market game:
$I=\{1,2...,n\}$ players
$S_i$ strategy space of each player $i\in I$
$u_i(s_i,s_{-i})=R_i(s_i)-C(s_i,s_{-i})$
There's only one type of resource. The resource is ...
- 11
3
votes
1
answer
175
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Regarding a proof of Tarski-Knaster's fixed point theorem
I'm trying to prove the following theorem,
Let $(X, \leq)$ be a complete lattice. If $\phi:X \to X$ is order preserving, it has a fixed point.
So far, I've done the following:
I've defined the ...
- 17.2k
1
vote
1
answer
197
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Simultaneous Least/Greatest Fixed Points
I have two functions $f : V \times S \to V$ and $g : V \times S \to S$ where $V$ and $S$ are complete lattices and $v \subseteq v' \wedge s \subseteq s'$ implies $f(v, s) \subseteq f(v', s') \wedge g(...
- 113
1
vote
1
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182
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Interpretation of Tarski's fixed point theorem
I have a question regarding Tarski's fixed point theorem. Consider the set $$\mathcal{X}:={0,1}^2:=\{(1,1), (0,1), (1,0), (0,0)\}$$
Consider the vector $X:=(X_1,X_2)$ taking value in $\mathcal{X}$.
...
- 266
1
vote
1
answer
119
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Common fixed point of commuting monotonic functions
Let $P$ be a chain-complete poset with a least element, and let $f_1,f_2,\ldots,f_n$ be order-preserving maps $P\to P$ such that $\forall i,j: f_i \circ f_j = f_j\circ f_i$.
Claim. The functions $f_1,...
- 1,293
0
votes
1
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350
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Poset where every monotonic function has a least fixed point
Let $P$ be a poset such that every order-preserving map $f:P\to P$ has a least fixed point. Must $P$ be chain-complete?
- 1,293
0
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1
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282
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Chain-complete and least element iff every order-preserving map has least fixed point
Let $P$ be a poset. I want to show the following are equivalent.
$P$ is chain-complete and it has a least element.
For every order-preserving map $f:P\to P$, the set $P_f$ of fixed points of $f$ has ...
- 1,293