All Questions
Tagged with lattice-orders boolean-algebra
120
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Finite distributive lattices and finite abelian monoids
A structure of semilattice over T is the same thing than a structure of finite abelian monoid such that $\forall t \in T$, $t² = t$.
Given a semilattice T, we get an abelian monoid by defining $a.b$ =...
- 302
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2
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162
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Sublattices of rank n of the Boolean algebra and partial orders
Let $f(n)$ be the number of sub lattices of rank n the Boolean algebra $B_n$.
I want to show that $f(n)$ is also the number of partial orders of $P$ on $[𝑛]$.
I have read this question from Counting ...
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Generalized boolean algebra structure on connected subset of euclidean space
This is a curiosity question that I've been grappling with as I've been reading more about lattice theory:
Is it possible to endow some connected subset of $\mathbb{R}^n$ with a generalized boolean ...
- 107
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59
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Embeddings of finite boolean algebras
I have finite boolean algebras $B_1, B_2$ and an injective homomorphism $e : B_1 \hookrightarrow B_2$ between them. I'd like to know whether the following fact is true:
Does there exist for every $y \...
- 485
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The Boolean lattice of a Boolean ring
I am proving that a Boolean Ring is also a Boolean Lattice.
I defined $\leq$ as $x\leq y$ when $xy=x$. The supremum is $a+b+ab$, and infimum is $ab$. The Max element is $1$, Min is $0$.
I proved that $...
- 319
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How is it the case that: Any complete lattice is a Boolean algebra.
In the book “A Functorial Model Theory” by Nourani (pg152), it is stated that
However, I didn’t understand what does he mean? Because a complete lattice is not even necessarily distributive whereas ...
- 375
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Is there a general poset representation that specializes to power set lattices in case of finite boolean algebras?
I read here that every finite, complemented, distributive lattice is isomorphic to a power set lattice.
Is there a general order preserving mapping from a poset $P$ to a set inclusion poset $S$, such ...
- 133
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Left adjoint to the inclusion of Boolean algebras into distributive lattices
Let $\mathbf{Boole}$ be the category of Boolean algebras.
Let $\mathbf{BDL}$ be the category of bounded distributive lattices.
There is a fully faithful functor ${\mathbf{Boole} \rightarrow \mathbf{...
- 279
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1
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73
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Symmetric relations form a CABA
Fix a set X and consider the collection of all symetric relations on it. I also assume that the empty relation is by definiyion symmetric. Well, it is true that the above collection forms a complete ...
- 5,192
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Is every Heyting algebra a sublattice of a Boolean algebra?
From what I can tell, every lattice is a sublattice of a lattice with unique complements (Dilworth). A Heyting algebra is a distributive lattice. The only remaining step, then, would be to know ...
- 97
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A Boolean algebra of functions
Consider a complete Boolean algebra $([A \to B], \leq_{[A \to B]})$ whose carrier set is the class of functions from the set $A$ to the set $B = \{1, 0 \}$, where $1$ and $0$ represent truth and ...
- 1,413
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76
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Are there complete Boolean algebras with no non-trivial complete endomorphisms?
I have read about the existence of rigid complete Boolean algebras that have no non-trivial automorphisms, and endo-rigid Boolean algebras that have only certain kinds of endomorphisms. So I was ...
- 1,297
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If $A\subset A'$ are Boolean algebras, can the Stone space of $A$ be embedded in the Stone space of $A'$?
I think the following is true but I can't find a reference. If $A$ and $A'$ are Boolean algebras, and $A \subset A'$, then $S_A$ can be embedded in $S_{A'}$, where $S_A$ and $S_{A'}$ are the ...
- 579
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Closure operators on powerset lattices generated by Galois connections from relations
In the book "Residuated Lattices: An Algebraic Glimpse at Substructural Logics" by Galatos, Jipsen, Kowalski and Ono they have this result (Lemma 3.8(2) page 147)
If $\gamma$ is a closure ...
- 531
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Completeness of a quantifier-free axiomatization of Boolean algebra using partial-ordering
Consider the following set of axioms:
$a \leq a$
$(a \leq b \mathrel{\&} b \leq a) \Rightarrow a = b$
$(a \leq b \mathrel{\&} b \leq c) \Rightarrow a \leq c$
$0 \leq a \mathrel{\&} a \leq ...
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