All Questions
Tagged with universal-algebra order-theory
22 questions
5
votes
1
answer
376
views
Kuratowski's 14 theorem and universal algebra
For a tuple of functions $\overline{p}$ on a set $Y$, let $cl_{\overline{p}}$ be the associated closure operation: $cl_{\overline{p}}(Z)$ is the smallest subset of $Y$ containing $Z$ and closed under ...
- 21.2k
3
votes
0
answers
176
views
Can the set of parafinite congruences be descriptive-set-theoretically complicated?
Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...
- 21.2k
1
vote
1
answer
142
views
Congruences that aren't "finite from above," take 2: semigroups
This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
- 21.2k
5
votes
3
answers
542
views
Congruences that aren't "finite from above"
Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...
- 21.2k
3
votes
1
answer
152
views
Posets of equational theories of "bad quotients"
This is a follow-up to an older question of mine:
Suppose $\mathfrak{A}=(A;...)$ is an algebra (in the sense of universal algebra) and $E$ is an equivalence relation - not necessarily a congruence - ...
- 21.2k
1
vote
1
answer
83
views
Sizes of linearly ordered subalgebras of powers
On the grounds that I'm currently teaching a linear algebra class and I enjoy making my students furious, let a linear algebra be an algebra $\mathcal{A}$ in the sense of universal algebra equipped ...
- 21.2k
8
votes
1
answer
356
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
3
votes
0
answers
133
views
Is there an ordered algebra analogue of the HSP theorem?
For an algebraic signature (= set of function symbols) $\Sigma$, say that an ordered $\Sigma$-algebra is a pair $\mathfrak{A}=(\mathcal{A};\le)$ where $\mathcal{A}$ is a $\Sigma$-algebra in the sense ...
- 21.2k
3
votes
1
answer
121
views
When does a clone on a two-element set have almost abelian symmetry groups?
Say that a clone (in the sense of universal algebra) $\mathfrak{C}$ has almost abelian symmetry groups (= aasg) iff for each function $f(x_1,...,x_n)\in\mathfrak{C}$ there is an abelian subgroup $A\...
- 21.2k
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 $x = 1$.
...
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
3
votes
0
answers
589
views
Braided lobsters
If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...
2
votes
1
answer
302
views
Name for this algebraic structure?
I've found myself looking at a structure $\mathbb{M}$ whose important properties are:
$\mathbb{M}$ is a discretely ordered additive monoid.
$\mathbb{M}$ has a least element, and this least element is ...
- 10.1k
8
votes
2
answers
483
views
Posets obtained from a semigroup by the definition $x \leq y \iff x \cdot y = x$
A po-groupoid is a groupoid $\langle A,\cdot\rangle $ such that the relation
defined by
$$
x \leq y \text{ if and only if } x \cdot y = x
$$
is a partial order on $A$, the order related to $\langle ...
- 1,124
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
3
votes
0
answers
119
views
Generalizing disjointness
The following definition generalizes set-theoretic disjointess:
Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...
- 3,793
1
vote
0
answers
78
views
Looking for a uniform explanation of algebras with canonical generators.
Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a ...
- 1,271
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
2
votes
2
answers
243
views
Of what kind of complemented bounded poset are the structures in my quasi-variety?
I feel that my question is very basic, but, somewhat suprisingly, nobody was able to give me an answer so far:
Let
$\mathbf{M} := \langle \{ 0,1 \}, 0, 1, \leq, \neg \rangle$
be the structure with ...
- 1,498
3
votes
2
answers
269
views
What do you call a lattice whose meet operation preserves disjointness of subsets?
To make my question more precise and compact (and probably more intuitive), let me define the following:
A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x \...
- 205
7
votes
1
answer
732
views
Does ⬦ generate all De Morgan algebras?
(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
- 1,119
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