All Questions
Tagged with model-theory universal-algebra
63
questions
1
vote
0
answers
41
views
Comparing semiring of formulas and Lindenbaum algebra
This is motivationally related to an earlier question of mine.
Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows:
Elements of $\widehat{D}(T)$ are equivalence ...
- 20.7k
2
votes
0
answers
127
views
On "necessary connectives" in a structure
Given a clone $\mathcal{C}$ over $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with connectives from $\mathcal{C}$ in place of the usual Booleans. Given a clone $...
- 20.7k
3
votes
0
answers
93
views
Are "equi-expressivity" relations always congruences on Post's lattice?
Given a clone $\mathcal{C}$ over the set $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with (symbols corresponding to) elements of $\mathcal{C}$ replacing the ...
- 20.7k
8
votes
0
answers
131
views
What is this quotient of the free product?
Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
- 20.7k
3
votes
0
answers
168
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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 ...
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1
vote
1
answer
142
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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 ...
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5
votes
3
answers
533
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 ...
- 20.7k
3
votes
1
answer
143
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 - ...
- 20.7k
1
vote
1
answer
81
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 ...
- 20.7k
9
votes
1
answer
295
views
Two notions of generalized quotient/substructure
Given a language $\Sigma$ and a $\Sigma$-algebra (in the sense of universal algebra) $\mathcal{A}=(A;\dotsc)$ and a function $f:A\rightarrow A$, let $\mathcal{A}_f$ be the $\Sigma$-algebra whose ...
- 20.7k
4
votes
0
answers
238
views
Cantor-Bernstein phenomena for structures (and a "moderate zigzag" property)
My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as ...
- 20.7k
1
vote
0
answers
176
views
Theorem constructing a mathematical structure from a set of internal isomorphisms
I am searching for information about a specific theorem mentioned in the book "Discriminator-algebras: algebraic representation and model theoretic properties" by Heinrich Werner. The ...
- 119
13
votes
0
answers
317
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When does HSP reduce to SPH?
This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...
- 17.8k
3
votes
0
answers
129
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 ...
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5
votes
0
answers
119
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Can the equational theory of commutative rings be "unpacked" from the equational theory of exponentiation?
Below, I'll use "$\approx$" for the equality symbol in an equation, as opposed to "actual" equality.
Suppose $\mathcal{V}$ is a variety (in the sense of universal algebra) in the ...
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