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Questions tagged [domain-theory]

Domain theory is a branch of order theory that studies partially ordered sets which are called domains. Do not use this tag for questions about the domain of a function.

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Decidability of a relation on Functional space!

Suppose I have this functional space $(D=\{ a\searrow b; a \in A , b \in B\}, \leqslant)$ (partial order relation on step functions!),also suppose that relation $\leqslant_1$ is decidable on $(A,\...
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Is the down-closure of a sub-dcpo a sub-dcpo?

Recall that a dcpo is a partially ordered set $(X,\leqslant)$ possessing suprema for all directed subsets. The Scott topology of a dcpo $(X,\leqslant)$ has as closed sets those sub-dcpos which are ...
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Is there a term for extensions of functions on discrete sets to continuous sets?

The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that $$\Pi(n) = (n)!$$ for all positive integers $n$. In other words,...
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Continuous Lattices - Confusing passage in Dana Scott's paper

In Dana Scott's paper called "Continuous Lattices" from 1971, I am lost in the following passage. This passage occurs in some exposition after his Proposition 2.4. Before going any deeper, however, ...
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Scott topology vs the topology generated by the complements of principal ideals

In Scott topology, it is easy to show that principal ideals are closed sets (and their complements are open). I suppose that in general case Scott topology is not necessarily generated by the ...
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Is this partially ordering of functions a dcpo? What are the compact elements?

Take the space, $F$, of functions $f: \mathbb{R} \rightarrow \mathbb{R}$. We know that every one of these can be seen as a vector in a suitable basis of functions (sine cosine, polynomials etc). Let ...
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What kinds of non-topological continuity concepts exist between partially-ordered sets

Beside Scott-continuity, what other kinds of continuity concepts are there between partially-ordered sets that can be stated without recourse to topological concepts?
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A Scott-like continuity between partially-ordered sets

Given two partially ordered sets $P$ and $Q$, a function $f : P → Q$ between them is Scott-continuous if it preserves all directed suprema, i.e. if for every directed subset $D$ of $P$ with supremum ...
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Maximal domain of function and its derivative if both have Lebesgue > 0

Consider $f(x) = \ln(x), \sqrt{x}, |x|$ and $\lfloor x \rfloor$. Based on these, I have the following conjecture. Please prove/disprove. The maximal domain of $f$ and its derivative differ only by ...
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The Domain (Co)Monad

Many different kinds of data structures can be captured as Monads. Lists and trees are two good examples. A domain (dcpo) is like a tree, with extra axioms. Definition. A directed subset of a ...
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how to translate from one “domain of a function” to anothe

I have these values, all these values are in the range between $0$ and $1 (0, +1)$. $0.3$ $0.5$ $0.8$ $0.9$ I want to change (translate) these values to a new range between $-1$ and $+1$ $(-1, +1)$, ...
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How is a Scott domain Cartesian-closed?

I read the following excerpt from nLab but I need further explanation: The problem Scott solved is to faithfully model untyped lambda calculus; in categorical terms, the basic problem is to ...
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Fractional ideals: Is $D$ included in $I'I$?

Let $D$ be a domain and let $F$ be its field of fractions. If $I$ is fractional, we define $I'$ to be the set of elements $c \in F$ such that $cI$ is included in $D$. It is clear that if $I$ is ...
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1answer
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Find the interval of $c$ such that the rational function $\frac{x^2+2x+c}{x^2+4x+3c}$ takes all real values

Find the interval of $c$ such that the rational function $$f(x)=\frac{x^2+2x+c}{x^2+4x+3c}$$ takes all real values $\forall$ $x\in D_f$ I tried in the following way: Let $$y=\frac{x^2+2x+c}{x^2+4x+...
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Limits in a dcpo with the Scott topology

This is exercise 4.7.7 in Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq. Consider a dcpo $(X,\leq)$ and equip it with the Scott topology where the opens are the upward-closed sets ...
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1answer
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what is K-completeness? (Kelly completeness)?

I'm studing Domain theory and working on maximal point space problem. I couldn't find any book or sci-text about K-completness and D-completness!Can any body give me some help with this?
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Scott topology on $\mathcal{O}X$

In Wikipedya Scott-open set is defined as follows: Def. A subset $O$ of a partially ordered set $P$ is called Scott-open if It is an up set. All directed sets $D$ with supremum in $O$ have non-...
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Inverse domain problem

My task is to find the domain of inverse of : $f(x) = \dfrac{\cos e^x}{1-\cos e^x}$ . Now, if I am correct the inverse should be : $f^{-1}(x) = \ln\Bigl(\arccos\dfrac {x}{x+1}\Bigr)$ . If I solve ...
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Domain definition for $R(a) = \{(s_1,s_2), …\}$

If I have $a \in A$ and $s \in S$ and different function values $R(a)$ which, for instance, could be $$R(a_1) = \{ (s_1,s_2),(s_2,s_3),\ldots \}$$ What is the definition of the function $R$? I guess ...
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Find $b$ and $c$ for $f(x)=\log(-x^2+bx+c) $

I have a function $f(x)=\log(-x^{2}+bx+c) $ and the domain of $f$ is $(1,3)$. I have to use this fact to find the values of $b$ and $c$. I thought about solving the inequality $1 < \log(-x^2+bx+c) &...
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Monotonicity, Continuity and Least fix point of function

I am learning semantics of language Haskell and there i come around this question. For answering this, I am thinking in this way: a) For a function to be monotonic, let's say I have 2 arguments: $f$ ...
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What is the relationship, if any, between maximal domains of $f$ and $f'(x)$?

(Edited question after some Calculus review) Let $D$ be the maximal domain of function $f$. Let $D^d$ be the largest subset of $D$ where $f'(x)$ is defined. Maximal domain of $f(x) = \ln(x)$ is $D_1 ...
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How would you sketch this subset of $\mathbb{C}$?

How would you sketch the subset $\{z \in \mathbb{C} : |e^{z}| <1 \}$ of $\mathbb{C}$? Is there a general method for this kind of problem? Also, how is it possible to tell whether a subset of $\...
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What is the domain of $f(x,y) = xy/\sqrt{x^2+y^2}$

I think the domain is all real except $(0,0)$, because this point anulates the denominator, but what about with the fact that the limit exists when $(x,y)$ tends to $(0,0)$?
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Why is the union of two closed sets again closed in the Scott topology?

I want to figure out why the Scott topology really forms a topology. In particular, I want to find out why the union of two closed sets is again closed. So say we have a partial order $P = (P, ≤)$. A ...
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$\mathfrak{ap\subsetneq a}$ , with $\mathfrak{a,p}$ ideals

Let be $D$ a domain, $\mathfrak{a,b,p}\subsetneq D$ ideals with $\mathfrak{ab}=\lambda D\, , \lambda\in D\setminus\{0\}$ and $\mathfrak{p}$ prime maximal. Show that $\mathfrak{ap\subsetneq a}$ Is ...
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What really is codomain?

Pretty simple: Is the codomain of a matrix simply the number of rows, or is it the rank? (The linearly independent rows?) Maybe I can ask about some examples. Here are the questions and my answers.
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$\mathfrak{a\subseteq b}$ and $\mathfrak{bc}=\lambda D$ then there is an ideal $\mathfrak{b}'$ such $\mathfrak{a=bb'}$

Let be $D$ a commutative domain, $\mathfrak{a,b,c}\subseteq D$ ideals. Show that: if $\mathfrak{a\subseteq b}$ and $\mathfrak{bc}=\lambda D$ then there is an ideal $\mathfrak{b}'$ such $\mathfrak{...
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If $A\subset \lambda D$ then $\exists A'$ such $A=\lambda A'$

Let be $D$ a commutative domain, $\lambda\in D\setminus \{0\}$ and $A\subset D$ an ideal. Then if $A\subset \lambda D$ then exists $A'\subset D$ such $A=\lambda A'$ my work: If $A=\lambda D$ since ...
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every set of non compact elements has a maximum element

Let $P$ be a poset. An element $x$ of $P$ is compact if for every directed set $D\subseteq D$, $x\leq \sup P$ implies $x\in {\downarrow\!\! D}$. Is it true if the set of compact elements of $P$, ...
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Why a function from a product of $\omega$-complete partial order is continuous?

In “The Formal Semantics of Programming Languages” by Glynn Winskel, there is Lemma 8.10 in “8. Introduction to domain theory/3. Constructions on cpo's/2. Finite products” which is promised to be “...
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Equivalence of categories of directed complete posets

In the book ``Domains and Lambda-Calculi'' by Amadio and Curien, there is the following exercise: Define an equivalence between the category of partial morphisms generated by $(\mathcal{M}_S, \textbf{...