All Questions
Tagged with universal-algebra lattice-theory
30 questions
2
votes
1
answer
118
views
Reference request for a proof of the fact that every congruence-permutable variety is semidegenerate
Given an algebra $\mathbf{A}$, a pair of congruences
$ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when
$ \alpha \circ \beta =\beta \circ \alpha$, and an algebra
$\mathbf{A}$ is called ...
- 21
4
votes
1
answer
254
views
Are there atoms in the lattice of intermediate logics?
A few days ago I stumbled upon this question on MS. The question is: Does the lattice of intermediate logics have an atom, i.e. an element that is strictly stronger than IPC but not strictly stronger ...
- 143
2
votes
1
answer
213
views
Parametrization of topological algebraic objects
There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
- 5,529
11
votes
1
answer
343
views
Lattices of clones: is 4 worse than 3?
Let $\mathscr{C}_n$ be the lattice of clones on the $n$-element set $\{1,...,n\}$. $\mathscr{C}_2$ is complicated but countable, but $\mathscr{C}_3$ (and all higher lattices) is of size continuum.
...
- 21.2k
10
votes
1
answer
676
views
Are modular lattices shallow?
Let $A$ be a universal algebra with finitely many finitary operations. Write $F_n$ for the $n$-ary operations.
We define the affine maps on $A$ inductively: $\eta \mapsto \eta$ and $\eta \mapsto c$ ...
- 6,652
0
votes
0
answers
101
views
Is a principal filter in a free Heyting algebra a projective Heyting algebra?
A Heyting algebra is a bounded distributive lattice $(L,\vee,\wedge,0,1)$ together with a binary operation $\rightarrow$ called implication or relative pseudocomplementation with the property that, ...
- 1,644
4
votes
1
answer
159
views
How large must algebras with a given congruence lattice be?
This is a follow-up to a recent question of mine:
For $n\in\mathbb{N}$ let $C(n)$ be the smallest $k$ such that every bounded lattice with cardinality $\le n$ which is isomorphic to the congruence ...
- 21.2k
8
votes
1
answer
357
views
Example of trickiness of finite lattice representation problem?
I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...
- 21.2k
8
votes
1
answer
787
views
A new and subtle order-theoretic fixed point theorem
Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...
- 8,481
7
votes
1
answer
193
views
Free median algebras and maximal linked systems
$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
- 63.9k
0
votes
1
answer
129
views
Ordering preserved by an inverse frame homomorphism
Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames):
Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$.
Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$.
...
23
votes
1
answer
970
views
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 ...
- 15.9k
18
votes
1
answer
2k
views
Uses of Zorn's Lemma when the thing is actually unique
There is a revised version, which I might substitute for this one, but I would like to keep this as evidence of priority for the "special condition".
Are there uses of the sledgehammer Zorn'...
- 8,481
3
votes
2
answers
124
views
Explicit lifting characterization of complete lattices among posets?
It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property ...
- 63.9k
1
vote
0
answers
48
views
Equivalence relations: Cosimplicial semilattice?
For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join
$$R\vee R' := \langle R\cup R'\rangle,$$...
- 1,623
0
votes
1
answer
94
views
What is the method to relax or weakening a structure "ripping off" from it all its identity elements?
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with ...
- 1
6
votes
2
answers
493
views
Finite lattice representation problem checking
[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem ...
3
votes
0
answers
67
views
Word problem for finitely presented bounded lattices
There is a solution to the word problem for finitely presented (non-bounded) lattices, as well as a solution to the word problem for free bounded lattices. I am assuming that there is a solution to ...
- 203
12
votes
2
answers
831
views
What is known about ideal and divisibility lattices of GCD domains and their generalizations?
The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...
- 439
19
votes
4
answers
1k
views
Representation theorem for modular lattices?
Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices?
For example, I ...
- 63.1k
13
votes
3
answers
1k
views
About a construction of Borel $\sigma$-algebra associated to a lattice
Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$).
Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i....
- 4,165
5
votes
1
answer
264
views
When are the congruence lattices nicer?
This is a purely idle question, but one I'm increasingly interested the more thought I put into it:
For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...
- 21.2k
5
votes
2
answers
355
views
Finite lattices whose number of join-irreducibles does not exceed its height
In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements.
Briefly, this follows by arranging the subposet $J(...
- 1,271
6
votes
1
answer
676
views
Generalizations of Birkhoff's HSP Theorem
Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and ...
- 1,307
2
votes
1
answer
230
views
Distributive lattice embedding into a finite lattice
Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice.
If $\iota$ preserves all meets and joins one can show that $...
- 1,271
0
votes
2
answers
214
views
Direct limit of lattice-ordered groups
In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have ...
- 173
5
votes
1
answer
298
views
Algebras with supremum-founded subalgebra lattice
I am interested in algebras whose subalgebra lattice is supremum-founded. Let us call those algebras small.
A complete lattice $(L, \leq)$ is called supremum-founded, if for any two elements $x < ...
- 1,498
22
votes
0
answers
1k
views
Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?
If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
- 1,241
8
votes
1
answer
1k
views
Lattice-ordered commutative monoids
By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...
- 44.4k
5
votes
2
answers
562
views
Is every lattice the fixed-point set of an order-preserving endomorphism of ⋄^n?
(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
Let $\diamond$ be the 4 element lattice
τ
/ \
i j
\ /
f
Is every lattice isomorphic to the fixed point lattice of some order-...
- 1,119