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Results tagged with lo.logic
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user 75735
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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
14
votes
Accepted
Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
Here is a near miss: Let $\mathcal{V}$ be the variety of $\mathbb Z_2$-sets. These may be thought of as algebras …
- 14.6k
10
votes
Examples of natural algebraic irreflexive relations
Are there any other interesting examples of natural algebraic irreflexive relations?
Let $\mathcal{V}$ be an equationally definable class of algebras
in a language that has at least two distinct const …
- 14.6k
7
votes
What is a "general" relation algebra?
There is one natural candidate that leaps to mind, namely the construal of $2^G$ as a relation algebra for each group $G$ $\ldots$
This construction will not answer your question, since each relation …
- 14.6k
14
votes
Exponentials of truth values
Is there something deeper behind this analogy?
If you have a bijection $\beta\colon X\to Y$, then to every operation on $X$ there is a ($\beta$-)conjugate operation on $Y$. Namely, if $f\colon X^k\to …
- 14.6k
7
votes
Additive, multiplicative, and Dedekind infiniteness in ${\sf (ZF)}$
But a theorem of Sageev says that $\ldots$
I don't want to alter Andreas's answer, but I want to add a reference to one of the results he mentions.
Sageev, Gershon
An independence result concerning th …
- 14.6k
4
votes
Accepted
Posets of equational theories of "bad quotients"
Let $\mathbb{R}=(\{\textrm{real numbers}\};0,1,+,\times)$. Is there an equivalence relation $E$ on $\mathbb{R}$ such that $\mathbb{P}_\mathbb{R}(E)$ is not upwards-directed?
I will assume that the ord …
- 14.6k
4
votes
Accepted
Classes of algebras axiomatizable by special formulas; and free objects
Question 1:
Given a class of algebras
$\mathcal{K}$,
what conditions on $\mathcal{K}$
imply that it can be axiomatized by a class of
$\mathcal{L}$-formulas
of a special type (such as an identity or qu …
- 14.6k
7
votes
Accepted
Two notions of generalized quotient/substructure
Let me copy Definition 4.1 of
Libor Barto, Jakub Oprsal, Michael Pinsker
The wonderland of reflections
Israel Journal of Mathematics 223 (2018), 363-398
Defn. 4.1
Let $\mathbf{A}$ be an algebra with s …
- 14.6k
2
votes
Invariant theory in universal algebra
Refinement of Questions 1 and 2: Is there any general result in Universal algebra (or Model theory) that is relevant for the purposes of these questions?
I think that this is too general of a question …
- 14.6k
8
votes
Lattices of clones: is 4 worse than 3?
This is not an answer, but a long comment. I want to post references that are relevant to a question raised in the comments, namely
(Wojowu) Is it possible that $\mathscr{C}_3$ has some weak universa …
- 14.6k
3
votes
Accepted
How large must algebras with a given congruence lattice be?
What do we know about the growth rate of $C(n)$?
We know the exact value of $C(n)$ if, in its definition,
we restrict to the class of distributive lattices.
Otherwise we only have partial results. Let …
- 14.6k
2
votes
Equational theories determined by "identities without variables"
This question was edited on March 12, 2023 and I was asked to comment on the edited form. Let me copy the essential part of the new question:
I believe $\ldots$ that a variety $\mathscr V$ is of the a …
- 14.6k
8
votes
Accepted
Example of trickiness of finite lattice representation problem?
$M_4$, the modular lattice of height two with four atoms is an example.
$M_4$ arises as the subgroup lattice of the symmetric group on $3$ letters, hence it arises as the congruence lattice of a regul …
- 14.6k
14
votes
Accepted
Is the equational theory of this "orthocentrish" algebra finitely based?
This algebra is finitely based.
In fact, if you choose any bijection from $\{a,b,c,d\}$ to $\mathbb Z_2\times \mathbb Z_2$, then you can transport the operation $F(x,y,z)$ to $\mathbb Z_2\times \mathb …
- 14.6k