# 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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### Finite lattices that are Koszul

Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$. It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
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### Book and article recommendations with the purpose of studying the intersection between probability theory and lattice theory

Lately, I have been studying probability theory and lattice theory separately and I would like to investigate ideas which relate both subjects together. Having said that, I would like to know if ...
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### Is $(\omega+1)^\omega/{\cal U}$ complete for ${\cal U}$ free ultrafilter?

Let ${\cal U}$ be a free ultrafilter on $\omega$. Is the linearly ordered set $(\omega+1)^\omega/{\cal U}$ complete?
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### Upper bound of rank of a matrix

Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the ...
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### Constructive lattice completion

The Dedekind–MacNeille Completion is the generalized way of completing an arbitrary lattice $L$. We will call $C$ the Dedekind–MacNeille completion of $L$ (I will not go into the details of the ...
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### Dedekind-MacNeille completion of ${\cal P}(\omega)/({\rm fin})$

Let ${\cal P}(\omega)/({\rm fin})$ be the quotient of the Boolean algebra ${\cal P}(\omega)$ where two sets are considered to be equivalent if they differ by a finite number of elements. It turns out ...
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### Medians in lattice theory

John Isbell defined a notion of a median algebra (although the original idea is due to Birkhoff). A median algebra is a set $S$ with a ternary operation $[x,y,z]$ satisfying a set of axioms. The ...
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### Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$

I am interested in the poset of all ideals of the local ring $$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$ $n=1$ is trivial. $n=2$ takes little work and it is shown below....
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### Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
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### Which Boolean lattices have a left-to-right symmetric drawing?

This question is inspired by a similar MSE question about partition lattices. Question: Which finite Boolean lattices have a symmetric drawing on the 2D plane? By a symmetric drawing of a lattice, I ...
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### "Pseudo-Boolean" lattice (almost every element has several complements)

If $(L,\leq)$ is a lattice with bottom element $0$ and top element $1$ and $x\in L$ we say that $y$ is a complement of $x$ if $x\vee y = 1$ and $x\wedge y = 0$. Is there a lattice $(L,\leq)$ with more ...
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### On free lattices

Free distributive lattices on a finite set exist and are finite, while free modular lattices on a finite set exist but are not finite when the set has at least 4 elements. Question: Is there a class (...
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### Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is ...
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### A meet-semilattice with top element that is not a lattice?

I am reading Francis Borceux’s “Handbook of Categorical Algebra I” and on page 135 it says In particular a finite version of 4.2.5 does not hold: a finitely complete and well-powered category ...
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### Periods of Coxeter transformation associated to root posets

$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra. Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix ...
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### Classification of posets that are quotient posets of the Boolean lattice

Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics. $B_n/G$ for a subgroup $G$ of the ...
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