Questions tagged [lattice-orders]
Lattices are partially ordered sets such that a least upper bound and a greatest lower bound can be found for any subset consisting two elements. Lattice theory is an important subfield of order theory.
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Can a free complete lattice on three generators exist in $\mathsf{NFU}$?
Also asked at MO.
It's a fun exercise to show in $\mathsf{ZF}$ that "the free complete lattice on $3$ generators" doesn't actually exist. The punchline, unsurprisingly, is size: a putative ...
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The right way to motivate lattice theory in a combinatorics class
I am attending a course on combinatorics. I was asked to present Möbius functions on lattices for this course. I was trying to look for a simple non-trivial problem that illustrates the need for ...
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Is there such a thing as 'overtification' (dual to compactification)?
The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions.
Is there a process ...
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Relative chinese remainder theorem and the lattice of ideals
Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
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What is the universal property of the prime spectrum of a commutative rig?
Let $A$ be a commutative rig, i.e. a commutative monoid equipped with a unital associative commutative bilinear multiplication and let $L$ be a distributive lattice. For the purposes of this question, ...
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Showing that poset of set of supports of a vector space is semimodular
Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by $\text{supp}(...
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Can the LCT and MCT for Lebesgue integrable functions be viewed as a lattice completeness result?
The set of Lebesgue integrable functions form a lattice under pointwise min and max (also more generally for R, Henstock-Kurzweil integrable functions with an upper or lower bound form a lattice as ...
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When is a commutative monoid with its natural order "continuous"?
Let $A$ be a commutative monoid and define its natural preorder by:
$$x\leq y :\Leftrightarrow \exists k\in A: x + k = y$$
Assume $\leq$ is antisymmetric.
What kind of "obvious" properties can ...
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Prove $(\mathbb Z \times \mathbb Z, \Sigma)$ to be a partial order and tell if its subset $T'$ is a lattice
Let $T = (\mathbb Z\times\mathbb Z, \Sigma) $ be defined as follows:
$$\begin{aligned} (a,b) \text{ } \Sigma \text { } (c,d) \Leftrightarrow (a,b) = (c,d) \text{ or } a^2b^2<c^2d^2\end{aligned}$$
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Improvement on the concept of separating families for the union-closed sets conjecture?
The union-closed sets conjecture states the following. Let $F$ be a finite family of finite sets that is union-closed and let $\cup (F)$ be the union of all sets in $F$. Then we can find an element in ...
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Lattices/Topology and the Stone Duality
For some context I have some partial understanding of lattices and an intermediate understanding of topology.
I at some point in the past week started thinking about a funny way to view a topology on ...
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Existence of paths obeying partial ordering
Consider a partially ordering on $\mathbb{R}^n$ that forms a lattice, with meet and join continuous w.r.t. the standard topology (i.e. a topological lattice). Can we choose a path $\gamma(t)$ with ...
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What is the distributive reflection of the lattice of ideals of a ring?
Let $A$ be a commutative ring and let $L$ be the set of ideals of $A$, partially ordered by inclusion. $L$ is a complete lattice but is not distributive in general. For instance, in the polynomial ...
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An order isomorphism between lattices is a lattice isomorphism
Suppose that $(X, \leq_X)$ and $(Y, \leq_Y)$ are ordered sets. Let $T:(X, \leq_X) \rightarrow (Y, \leq_Y)$ be an order isomorphism. Is it true that $T$ is a lattice isomorphism?
I have this question ...
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Books on pseudocomplemented lattices and Heyting algebras
I was wondering if anyone knows a good reference for pseudocomplemented lattices and/or Heyting algebras. Ideally, it should be something like Givant & Halmos's Introduction to Boolean Algebras, ...
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