Questions tagged [order-theory]
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712 questions
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Fractional ideals of maximal orders in quaternion algebras
Let $D$ be a skew field that is central and finite-dimensional over a number field $F$ (in particular: a quaternion algebra over $F$). Let $\Delta \subseteq D$ be a maximal $\mathcal{O}_{F}$-order. ...
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Fixed-point free shrinking bijection
Let $J\subseteq {\cal P}(\omega)$ be the collection of infinite subsets whose complement is also infinite.
Is there a fixed-point free bijection $\varphi:J\to J$ such that $\varphi(j)\subseteq j$ for ...
- 5,429
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Is every multiplicative lattice isomorphic to the lattice of ideals of some ring?
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 ...
- 11.9k
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Examples of structures that obey a "triangle inequality of tree paths"
Let $T$ be a connected acyclic simple graph, i.e. a tree. For all $x, y \in T$, we may define $[x, y) \subseteq T$ as the unique (non-overlapping) path from $x$ to $y$, including $x$ but excluding $y$....
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On the role of implicit orientation/choice in the definition of ordered pairs and Cartesian products
In ZF (or ZFC), ordered pairs are typically defined via standard set-theoretic encodings (e.g. Kuratowski pairs), and Cartesian products and relations are defined in terms of these ordered pairs.
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6
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1
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Complements of atoms
Let $L$ be a complete lattice. Usually the bottom element of $L$ is denoted by $0$, and the top element by $1$. We say that $a\in L$ is an atom if $[0,a] = \{0,a\}$, that is, there are no elements ...
- 53.9k
3
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Additively idempotent semirings that are not lattices
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 ...
- 16k
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Are complete linear ordered sets decomposable?
The starting point of this question is bof's classification in this comment of indecomposable ordinals. In particular, every complete well-ordering on more than $1$ point is decomposable. Also, the ...
- 55.6k
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Do the diamond chains have minimal blocking sets?
Let $A$ be a finite set of statements and $R$ is the set of all implications between them. Let $K$ be the complete relation of all arrows between elements of $A$. If there are no equivalent statements ...
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Does PFA imply that any $\aleph_1$-dense subsets of the real line are order-isomorphic via a nonexpansive bijection?
According to Theorem 6.9 in Baumgartner's survey "Applications of the Proper Forcing Axiom", under PFA, any two $\aleph_1$-dense subsets of the real line are order-isomorphic.
On the other ...
- 45.5k
4
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Checking topologies of orders and suborders
Surprisingly, there is an order-embedding $\iota:[0,1]\to ([0,1]\setminus \mathbb{Q})$, as nicely described by Emil Jeřábek in this answer. So $[0,1] \cong S:=\text{im}(\iota)\subseteq [0,1]\setminus \...
- 1,062
3
votes
1
answer
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Non-topological argument for the non-existence of an order-embedding $\iota: [0,1]\to ([0,1]\setminus \mathbb{Q})$
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]\...
- 51.6k
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Order-preserving injection $\iota:[0,1]\to X$ for large $X\subseteq [0,1]$ [duplicate]
Inspired by this older question: If $X\subseteq [0,1]$ such that $|X|=2^{\aleph_0}$, is there necessarily an order-preserving injection $\iota:[0,1]\to X$?
- 4,813
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Cutsets and antichains in ${\cal P}(\omega)$
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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"countable linear orders" = "free category with pointed endofunctor and $\omega$-shaped colimits"?
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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