# Questions tagged [lattice-theory]

The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.

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### The lattice of analogues of Robinson's $Q$

This question was asked and bountied at MSE without response. Call a sentence $\varphi$ in the language of arithmetic $Q$-like iff $\mathbb{N}\models\varphi$ and $\{\varphi\}$ is essentially ...
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### Structure of connected (finite undirected) graphs with respect to homomorphism

After the appropriate quotient, (finite undirected) graphs with homomorphism form a lattice: the meet is the categorical product G x H, the join is the disjoint union G + H. This question concerns the ...
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### Which lattices are quotients of finite powerset lattices?

Let $S$ be a finite set, and let $2^S$ be its powerset, regarded as a lattice. Let $L$ be a quotient (in the category of lattices and maps which preserve $\top,\bot,\wedge,\vee$) of $S$. What can we ...
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### Lattices from quaternion algebras (MAGMA software)

I am studying the paper "Lattice Packing from Quaternion Algebras" from 2012 about the construction of ideal lattices. In Section 3.3 the authors construct very interesting examples of lattices using ...
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### What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?

Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d_T$ induced ...
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### Open sets on a Stone space

If $B$ is a Boolean algebra (possibly assumed complete), is there a standard name for the Heyting algebra (or frame) $L := \Omega(S(B))$ of open sets on the Stone space $S(B)$ of $B$, — or for the ...
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### Relative strength and propositional indistinguishability of non-distributive lattices

Consider the class of bounded non-distributive lattices $\mathbf{Mn}$ ($n\geqslant 3$). (from left to right: M3, M4, Mn) Now consider a propositional language over $\{\wedge,\vee,\neg\}$ with the ...
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### Finite posets for which all intervals are atomic

Let $P$ be a finite poset which is a lattice with $0,1 \in P$. An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$. $P$ is atomic if every element is a join of atoms and ...
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### What's concrete model for Coxeter complexes?

We know for every Coxeter system $(W,S)$ there is a Coxeter complex associated by its cosets of parabolic subgroups. In Wachs's note Poset Topology p.12-13 she mentioned for the Coxeter complex of ...
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### Distributive generators of a lattice

Given a finite lattice $L$, suppose $L$ is generated by a set $X$ such that the distributive law holds for all $a,b,c\in X$ i.e. $a\lor (b \wedge c) = (a\lor b)\wedge (a \lor c)$. Is $L$ ...
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### Is the intersection of Boolean sublattices a Boolean sublattice?

Let $L$ be a boolean lattice, $A$ and $B$ sublattices of $L$ that are themselves boolean lattices, and suppose that $I = A \cap B$ is nonempty. Is $I$ a boolean sublattice of $L$? Is it a ...
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### Fibers of the morphism from the free Heyting algebra to the free Boolean algebra

For $k\in\mathbb{N}$, let $H_k$ be the free Heyting algebra on $k$ variables $p_1,\ldots,p_k$ and $B_k$ be the free Boolean algebra on the same $k$ variables. Thus, $B_k$ has $2^{2^k}$ elements (...
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### The space of Borel function modulo comeager sets is Dedekind complete

Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...
The XYZ Theorem of Shepp  states that for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for any three ...