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Inspired by this older question: If $X\subseteq [0,1]$ such that $|X|=2^{\aleph_0}$, is there necessarily an order-preserving injection $\iota:[0,1]\to X$?
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Starting point. I was toying around with the following question: If $A, B\subseteq [0,1]$ with $A\cup B = [0,1]$, does $[0,1]$ order-embed into at least one of $A, B$? Quickly I focused on $A = [0,1]\...
Dominic van der Zypen's user avatar
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If $(P,\leq)$ is a poset, an antichain is a set $A\subseteq P$ such that for all $a\neq b\in A$ we have $a\not\leq b$ and $b\not\leq a$. A chain is a subset $C\subseteq P$ such that for all $c,d\in C$ ...
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It is possible to show that the category of finite linear orders (i.e. the index category used to define augmented semi-simplicial sets) is equivalent to the free category with a pointed endofunctor ...
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Suppose we have two measure spaces, $(\Omega, \mathscr{F}, \mu)$ and $(\Psi, \mathscr{G}, \nu)$, with $\mu(\Omega) = \nu(\Psi) < \infty$, and we consider the set of measure isomorphisms mod 0 ...
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Let $A=KQ/I$ be an acyclic quiver algebra with Cartan matrix $U$ and let $n$ be the number of vertices of $Q$. For example when $A=KP$ is the incidence algebra of a finite poset $P$, then $U$ is just ...
Mare's user avatar
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I am looking for examples of additively idempotent semirings (which are always join semilattices) that do not have an underlying lattice structure, i.e. either meets do not exist or exist outside the ...
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On a lattice $\mathcal{L}$, I have a submodular, monotone, real-valued function $\rho$. Submodularity means it satisfies the inequality $$\rho(x)+\rho(y)\ge \rho(x\wedge y)+\rho(x\vee y)$$ for all $x, ...
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Richard Laver finishes his seminal paper "On Fraïssé's order type conjecture", with: Finally, the question arises as to how the order types outside of $M$ behave under embeddability. For ...
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A chain ${\cal C}\subseteq {\cal P}(\omega)$ is a set such that for all $A, B\in {\cal C}$ we have $A\subseteq B$ or $B\subseteq A$. Using Zorn's Lemma one can show that every chain is contained in a ...
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An antichain in $\mathcal P(\omega)$ is a set $\mathcal A\subseteq \mathcal P(\omega)$ such that for all $A, B\in \mathcal A$ with $A\neq B$ we have $(A\setminus B)\neq \emptyset$ and $(B \setminus A)\...
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It is well-known that in $\mathsf{ZF}$, the Axiom of Choice and Well-ordering Theorem are equivalent. What is perhaps less well-known is that there is a "local" version of this equivalence. ...
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If $(P,\leq)$ is a partially ordered set, we say that $C\subseteq P$ is a chain if $a\leq b$ or $b\leq a$ for all $a,b\in C$. An antichain is a set $A\subseteq P$ with $a\not \leq b$ and $b\not\leq a$ ...
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If $H=(V, E)$ is a hypergraph, the its chromatic number $\chi(H)$ is the smallest non-empty cardinal $\kappa$ such that there is a map $c:V \to \kappa$ such that for every $e\in E$ containing more ...
Dominic van der Zypen's user avatar
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If $(P,\leq)$ is a poset, we say that $C\subseteq P$ is a chain if $a\leq b$ or $b\leq a$ for all $a, b\in C$. Moreover, $A \subseteq P$ is an antichain if $a\not\leq b$ and $b\not\leq a$ whenever $a,...
Dominic van der Zypen's user avatar
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Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a chain if for all $A, B\in {\cal C}$ we have $A\...
Dominic van der Zypen's user avatar
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Let $T_R$ be the theory of rings in the language $\{+,\cdot,-,0,1\}$. Let $A$ be the set of one-variable sentences which imply commutativity over $T_R$, and let $B$ be the set of one-variable ...
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Let $k\in \mathbb{N}_+$, let $\mathcal{P}_k$ denote the set of directed graphs obtained as Hasse diagrams of posets on $k$ vertices, and let $\mathcal{Dir}_k$ denote the set of connected directed ...
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I don't know anything about algebraic geometry. I was bored at work, reading nLab, and noticed that the Zariski topology and Scott topology are vaguely similar. Strictly $T_0$, and almost never ...
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$\def\Pfin{{\cal P}(\omega)/{\text{(fin)}}}$The projection $\pi:{\cal P}(\omega)\to \Pfin$ is clearly surjective and order-preserving. (The quotient $\Pfin$ is defined here.) Is there a surjective ...
Dominic van der Zypen's user avatar
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Let $L$ be a finite lattice. Say $L$ has the unique factorisation property, if every element $x$ of $L$ is the join of a unique antichain of join-irreducible elements (or maybe one should use join-...
Mare's user avatar
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Let $L$ be a finite graded lattice with rank function $r$. According to the definition on page 39 in the book "Combinatorial Theory" by Aigner, L is said to have the regularity property $(...
Mare's user avatar
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If $V$ is a set, we define $V^V$ to be the collection of all functions $f:V\to V$. If $(V_i, E_i)$ are simple, undirected graphs for $i=0,1$, that is $E\subseteq {V\choose 2}$, we say that a map $f:...
Dominic van der Zypen's user avatar
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Let $n$ be a natural number, and $\mathscr{P}$ be the set of all preorderings (i.e., reflexive and transitive relations) on $\{1,\ldots,n\}$. (Equivalently, we can defined $\mathscr{P}$ as the set of ...
Gro-Tsen's user avatar
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I am writing a paper in which I make use of nets (Moore-Smith sequences). The terminology I am using comes from Eric Schechter - Handbook of Analysis and its Foundations (1996). There are several ...
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A multiplicative lattice is a complete lattice $(L, \leq)$ that is endowed with an associative, commutative multiplication that distributes over arbitrary joins and has $1$, the top element of $L$, as ...
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I want to know whether a partial order with bounded suborders of suitably many isomorphism types exists. More precisely, let $\mathbb{P} = (P, \leq_{\mathbb{P}})$ be a partial order and $\kappa$ be a ...
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I feel the need to explain my background before diving into this soft question, for you to understand my position. During my undergraduate years, a Theoretical Computer Science professor asked me to ...
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My education is mostly in general topology, so forgive me if this is obvious for set theorists. I'm starting to learn more about hyper-real fields (I'm reading Super-real fields by Dales and Woodin), ...
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Motivation. Let $X$ be a non-empty set. If $\tau$ is the trivial topology $\{\varnothing, X\}$ or the discrete topology ${\cal P}(X)$, then every function $f:X\to X$ is continuous. For the topologies ...
Dominic van der Zypen's user avatar
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Let $C_n$ be a chain (totally ordered set) of $n$ elements. For each $x \in C_n$, denote by $\iota(x)$ the number of elements in $C_n$ that are less than $x$. Consider the set of order-preserving ...
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This question is originally from Math StackExchange, but I now think it would be more suited here. I have added the [order-theory] tag as I think people experienced with Dickson's lemma may be ...
C7X's user avatar
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Let $\kappa \geq \aleph_0$ be a cardinal, and let $\newcommand{\Top}{\text{Top}}\Top(\kappa)$ be the complete lattice of all topologies on $\kappa$. For all $\tau\in \Top(\kappa)$ we have $\...
Dominic van der Zypen's user avatar
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My question arose after learning about Aronszajn trees. An Aronszajn tree is a rooted tree of height $\omega_1$ but where all levels are countable and all branches have countable height. An Aronszajn ...
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I am working with a product lattice and trying to show that $$ O(\Sigma L_1) \times O(\Sigma L_2) $$ is a sublattice of $L_3$. I already know that $$ O(\Sigma L_1 \times \Sigma L_2) $$ is a sublattice ...
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Let $S$ be a simplicial complex, and let $(P,\leq)$ denote the partial ordering of simplicies ordered by inclusion. That is, a 0-simplex $x$ is smaller than every 1-simplex that it bounds, every 2-...
Joe's user avatar
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Let $(P,<)$ be a dense linear order, i.e., for any $x,y\in P$ with $x<y$, there exists $z\in P$ such that $x<z<y$. A map $f:P\to P$ is called progressive if for all $x\in P$, $x< f(x)$. ...
Ray's user avatar
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Let $x$ be an element in a complete partial order. What's the accepted terminology, if any, for the following? Any $y\ge x$ such there does not exist any $z\ge y$. A greatest upper bound for $x$, ...
PythonAteMyHamster's user avatar
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It is a well known fact that the sublocales of the locale $L$ are defined by idempoten $\wedge$-semilattice endomorphisms, known as nuclei. Each nucleus $j$ of a locale $L$ also defines a filter $\...
Nik Bren's user avatar
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I am basing some of my thesis on Introduction to Boolean Algebras by Givant and Halmos. My current goal is to leverage the countable chain condition to define conditional probability measures. In ...
P. Quinton's user avatar
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This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested: Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
David Gao's user avatar
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The following might be a somewhat esoteric question: Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
David Gao's user avatar
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If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
Dominic van der Zypen's user avatar
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For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...
Gro-Tsen's user avatar
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The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
Gro-Tsen's user avatar
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Let $(P,\leq)$ be a partially ordered set. Assume that $(P,\leq)$ is well-founded. Then the levels of the poset are well-defined: $L_{0}$ is the set of minimal elements in $P$, and whenever $\mu$ is ...
Dillon M's user avatar
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Consider an object $X$ in an abelian category $\mathcal{C}$. We define $\text{Idem}_{\mathcal{C}}(X)$ as the set of idempotent endomorphisms $a$ in $\text{End}_{\mathcal{C}}(X)$, meaning that $a \circ ...
Sebastien Palcoux's user avatar
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Background: Let $(D,\, {\leq})$ be a partial order. We consider sequences $a = (a_n)_{n \in \mathbb{N}}$ of elements of $D$. For two sequences $a$ and $b$, we lift $\leq$ (point-wise) to sequences by ...
blk's user avatar
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Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
Capybara's user avatar
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A finite discriminating family in a group $G$ is a finite set of non-identity elements such that every non-trivial normal subgroup of $G$ contains one of these elements. A $k$-marked group is a group ...
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