All Questions
Tagged with lattice-orders abstract-algebra
188
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Galois connections give rise to complete lattices
I am reading Introduction to Lattices and Order, second edition, by Davey and Priestly. On page 161, it says
Every Galois connection $(^\rhd,^\lhd)$ gives rise to a pair of closure operators, $^{\rhd\...
- 1,889
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If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate]
Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. ...
- 3,468
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Galois connections for contexts
I am reading Introduction to Lattices and Order, Second Edition by Davey and Priestly.
On page 158, it says that
It can be verified that, for $R\in \mathcal{R}$ and $T\in \mathcal{T}$,
$$R\subset R_{...
- 1,889
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Is the following object a lattice?
I'd like to check that the following object is a lattice, since I'm having trouble understanding what I read as a slight ambiguity in the definition I've got in front of me.
The set $\{\emptyset,\{1\},...
- 3,326
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Example of a residuated prelinear lattice that isn't linear
A residuated lattice is an algebra
$$(L,\land, \lor, \star,\Rightarrow,0,1)$$
with four binary operations and two constants such that
$(L,\land,\lor,0,1)$ is a lattice with the largest element 1 and ...
- 49
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80
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Relation between semiring structure of natural numbers and their hereditarily finite sets structure under bitwise operations
Some background
The natural numbers $\mathsf{Nat}=\{0,1,2,\dots\}$ has the structure of a semiring under addition and multiplication.
Write $\mathsf{HFS}$ for the set of all hereditarily finite sets (...
- 538
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How to show that lattice of subgroups D4 isn't modular lattice?
Here is a lattice of subgroups D4.
The lattice isn't modular iff there is a "pentagon" as a sublattice.
As we can see $\left \{ \rho_{0} \right\} - \left \{ \rho_{0}, \mu_1 \right\} - \left ...
- 125
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Is there any kind of distributive law for $c\cdot\max(a,b)$, allowing both signs of $c$?
Notation: For any two numbers $a$ and $b$, let the maximum be $a\sqcap b$, and let the minimum be $a\sqcup b$. (No, my symbols aren't upside-down. Compare this with floor notation; $\lfloor a\rfloor\...
- 5,726
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Contracting a segment in a Lattice
Studying lattices, I'm looking for the following construction:
Given a finite (hence bounded and complete) lattice $L$ and $a,b \in L$ such that $a\leq b$,
obtain a lattice $L'$ by "contracting&...
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Multiplicative lattice with $(a \land b)\ast(a\lor b)=a \ast b$
The natural numbers $\mathbb{N}$ carry two (order-theoretic) lattice structures: One, say $L_1$, is the division lattice (where the join is the least common multiple and the meet is the greatest ...
- 1,769
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77
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Algebraic structures with opposite subalgebra lattices and congruence lattices that are not groups with operators
Are there any algebraic structures where the congruence lattice and subalgebra lattice are opposites that do not look like groups with operators?
My motivation for this question is noticing that ...
- 11.9k
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Database of lattices and lattice properties
Are there any websites that are databases of lattices?
I'm also interested in databases distributed as libraries in a programming language or similar.
I'm studying a little bit of lattice theory on ...
- 11.9k
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42
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Is there an infinite version of the brute-force lattice congruence representation algorithm that succeeds?
I am interested in the congruence lattices of algebraic structures.
Is there a brute-force way to produce an infinite algebraic structure whose lattice of congruences is isomorphic to some given ...
- 11.9k
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How does one go about "finding" the algebra of a particular logic?
My question is this: If I know a deductive system for a logic is there a simple way to derive the appropriate algebra for that system?
For example: I start with the natural deduction system for ...
- 418
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Stone's criterion for distributive lattices
A lattice $L$ is distributive iff for every $a\ne b$ in $L$ there is a prime ideal $P\in\textrm{Spec}(L)$ s.t. either $a\in P\not\ni b$ or $b\in P\not\ni a.$
Right to left. Since $D:L\to2^{\textrm{...
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