26
votes
Accepted
Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?
Following François's suggestion, I ran alg to find a unital commutative semiring which fails to satisfy
$$
\forall x\, y\, z,\; x + z = y + z \land x \times z = y \times z \Rightarrow x = y.
\tag{1}
...
- 48.8k
17
votes
Are arbitrary nonempty intersections of principal filters principal?
Let $L$ be the lattice of finite and cofinite subsets of $\mathbb N$. Let $\mathcal F_n$ be the principal filter of elements of $L$ that contain the element $n$ (i.e., $\mathcal F_n$ contains the ...
- 14.6k
17
votes
Accepted
Are the Boolean algebras ${\cal P}(\omega)/(\text{fin})$ and ${\cal P}(\omega)/(\text{thin})$ isomorphic?
The answer is no.
In the Boolean algebra $P(\omega)/\text{Fin}$ every strictly descending sequence $A_0>A_1>A_2>\cdots$ has a nonzero lower bound, by the famous construction of Hausdorff. ...
- 236k
17
votes
What's the deal with De Morgan algebras and Kleene algebras?
There are a lot of questions bundled together here. I will give some references for some of the questions.
An early paper on these topics is:
Lattices with involution
J. A. Kalman
Trans. Amer. Math. ...
- 14.6k
16
votes
Accepted
Is it possible to completely embed complete Heyting algebras into upsets of a poset?
No, not in general: for instance, the real interval $([0,1],{\le})$, or any non-atomic complete Boolean algebra, does not have such an embedding. This follows from the following characterization:
...
- 47.1k
15
votes
Class of lattices that excludes $M_3$?
This is a (slightly edited) copy of the answer I posted on math.SE to the question What do we call a lattice that does not have a sublattice the shape of the diamond $M_3$?:
Let $\mathbf K$ be the ...
- 13.4k
14
votes
Accepted
Group structure for distributive lattices
No:
If it's natural, it should be invariant under the automorphism group of the original lattice.
let $X$ be the free distributive lattice on 2 generators $x,y$: it has 6 elements, $$0\quad<\quad x\...
- 63.9k
14
votes
Are arbitrary nonempty intersections of principal filters principal?
There are two good answers already; but I’ll add a little motivation for how one might find the way to them.
If $L$ is complete, then as you say, it’s easy to see that any intersection of principal ...
- 21.3k
14
votes
Accepted
Determining whether a lattice is the face lattice of a polytope - NP hard or undecidable?
There's no contradiction:
I don't know the correct complexity, but I recall hearing several times that it is at least as hard as NP.
It is not difficult to show (using Tarski's algorithm, as ...
- 1,356
14
votes
Accepted
Are modular lattices shallow?
Terminology.
If $f$ is a fundamental operation of an algebra $A$, then a polynomial of the form $f(c_1,\ldots,c_k,x,c_{k+2},\ldots,c_n)$ is often called a 'basic translation' or a '$1$-translation' ...
- 14.6k
13
votes
Accepted
Is this Wikipedia article linking to the wrong notion of coherent space
Yes, the notion of "coherent space" at the page you linked is completely unrelated to spectral spaces. The coherent spaces of that article were introduced by Jean-Yves Girard as a denotational ...
- 1,646
13
votes
Accepted
What is known about ideal and divisibility lattices of GCD domains and their generalizations?
Given a ring $R$, let us denote by $L(R)$ the lattice of two-sided ideals of $R$ for which the infimum and supremum are given by $\inf(I, J) = I \cap J$ and $\sup(I, J) = I + J$.
Such lattices are ...
- 7,893
13
votes
Accepted
A meet-semilattice with top element that is not a lattice?
It is well-known that a finite meet-semi-lattice with a maximum element is a lattice. The reason is that we can define $a \vee b := \wedge \{c\colon \textrm{$c$ is an upper bound for $a,b$}\}$, where ...
- 24.2k
13
votes
Accepted
Ultrafilter lemma for arbitrary lattice
It is equivalent to AC.
Consider any collection $A$ of nonempty sets, and let $\newcommand\P{\mathbb{P}}\P$ be the set of partial choice functions, so that $p\in\P$ if and only if $p$ is a partial ...
- 236k
13
votes
A note on orders in quaternion algebras
Two orders need not be isomorphic.
First of all, in number fields $K$ other than $\mathbf Q$ not all orders are isomorphic rings (even if they are isomorphic abelian groups): the full ring of integers ...
- 50.6k
13
votes
Whether an isotone bijection from a power set lattice to another sends singletons to singletons
Yes, such a mapping necessarily sends singletons to singletons.
Let $f\colon\mathcal P(S)\to\mathcal P(T)$ be a monotone bijection (or more generally, a surjective strictly monotone function). By ...
- 47.1k
13
votes
Accepted
Whether an isotone bijection from a power set lattice to another sends singletons to singletons
Here is a more general fact:
Proposition: Any monotone bijection $f: A \to B$ between two Boolean algebras is an isomorphism.
Claim: $f(0) = 0$ and $f(1) = 1$.
Proof of Claim: For all $a \in A$, $0 \...
- 2,800
13
votes
Accepted
The reals: a topological lattice in more than the obvious way?
These are the only ones.
Write $x \preceq y$ if and only if $x = x \wedge y$ if and only if $y = x \vee y$, and write $\leq$ for the usual partial order on $\mathbf R$. To avoid some case separations, ...
- 28.7k
12
votes
Accepted
Automorphisms of power set lattice mod finite
This is known as Rudin-Shelah problem. Note that, by Stone duality, this is equivalent to determine the self-homeomorphism group of the Stone-Cech boundary of $N$. Notably, consider the group induced ...
- 63.9k
12
votes
Accepted
Fibers of the morphism from the free Heyting algebra to the free Boolean algebra
$\let\eq\leftrightarrow$Notice that $\psi(A)=u$ iff $\vdash_\mathrm{CPC}A\eq u$ iff $\vdash_\mathrm{IPC}\neg\neg(A\eq u)$. (I will write just $\vdash$ for $\vdash_\mathrm{IPC}$.) Thus:
$\bot$ has a ...
- 47.1k
11
votes
Accepted
Are all monomorphisms in the category of bounded lattices regular?
Is there a monomorphism in $\mathbf{𝐋𝐚𝐭}_{01}$ that is not regular?
No, all monomorphisms in the variety $\mathbf{𝐋𝐚𝐭}_{01}$ are regular.
Reasoning: The variety $\mathbf{𝐋𝐚𝐭}_{01}$
has the ...
- 14.6k
11
votes
Accepted
Which lattices are quotients of finite powerset lattices?
The class of finite powerset lattices is closed under quotients, up to isomorphism. That is, the quotients are exactly the lattice reducts of finite Boolean algebras.
In particular, any quotient $L$ ...
- 47.1k
11
votes
Accepted
Is every homogeneous poset a lattice?
Counterexample. Let
$$P=\{(x,i)\in\mathbb Q\times\{0,1\}:0\le x\le1,\ x\ne i\}$$
be ordered so that
$$(x,i)\lt(x',i')\iff x\lt x'.$$
- 13.4k
10
votes
Accepted
Order-preserving surjection ${\mathbb N}^{\mathbb N}\to [0,\infty)$
Yes. Let us isomorphically identify the poset of functions $\omega \to \omega$ (under the pointwise order) with the set of functions $\omega \to \mathbb{N}_2 = \{n \in \mathbb{N}: n \geq 2\}$, again ...
- 53.3k
10
votes
Accepted
Does the lattice of all topologies embed into the lattice of $T_1$-topologies?
Is there an injective lattice homomorphism $\varphi: \text{Top}(\kappa)\to \text{Top}^{T_1}(\kappa)$?
The answer is Yes, there is such an embedding.
I will argue that if $\kappa$ is an
infinite ...
- 14.6k
10
votes
Accepted
A formula for a right adjoint in terms of a left
Just so that it is recorded here as an answer, here's the formula from the "naive" adjoint functor theorem that directly generalizes the one given in the post for posets:
$$f^\bullet(y) = \lim_{(x,\...
- 66.7k
10
votes
Class of lattices that excludes $M_3$?
As bof says, the class $\mathcal K$ of lattices omitting $\mathbf M_3$ as a sublattice is a quasivariety that is not a variety. As bof also says, this implies that $\mathcal K$ is closed under the ...
- 14.6k
10
votes
Lattices on classical combinatorial families
Lattices are prevalent when one deals with integer partitions.
Let me give a few examples with pictures, that I hope you will enjoy despite the poor quality due to bitmap conversion.
The dominance ...
- 1,444
10
votes
Accepted
A definition in poset theory
I recall seeing in various sources the terminology "cover preserving embedding" and "cover preserving subposet". Googling it now (https://www.google.com/search?q=poset+%22cover+...
- 16.9k
10
votes
Is a finite lattice determined by its Hasse diagram (as a graph)?
This is still far from true. Consider for example these lattices (just a random example):
- 24.2k
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