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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
16
votes
Accepted
Is the class of additive groups of rings axiomatizable?
The answer to the question in the title is No. You can prove this from the work of Wanda Szmielew on the elementary properties of abelian groups. This answer works for any kind of nonzero, bi-additive …
- 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
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
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
13
votes
What classes of groups can arise as "symmetry groups of terms"?
Let me edit this response in order to clarify what I am showing.
First, I will begin with an example: $t(x_1,x_2,x_3,x_4,x_5,x_6)
= ((((x_1x_2)x_3^{-1})x_3)x_4)x_4^{-1}$
is a group term. If you want …
- 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
9
votes
Topological universal algebra: what is a variety?
This is a long comment rather than a complete answer.
But before writing it let me insert that I don't agree that universal algebra is the study of varieties. (In my universe, universal algebra is syn …
- 14.6k
9
votes
Accepted
Is the union of a chain of $\kappa$-colorable subgraphs $\kappa$-colorable?
For a counterexample, let $G$ be the complete graph on the ordinal $\omega_1$, let the $W$'s be the countable ordinals, and let $\kappa=\aleph_0$.
- 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
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
7
votes
Accepted
Is the quasi-equational theory of groups the same as cancellative semigroups?
No.
Every group satisfies
$$
(xy\approx x'y')\wedge (zy\approx z'y')\wedge (zw\approx z'w')\to (xw\approx x'w')
$$
but this quasi-identity is not derivable from associativity + cancellativity. You …
- 14.6k
7
votes
Free algebras from model theory perspective
Here are some papers.
(1)
Baldwin, J. T.; Shelah, S.
The structure of saturated free algebras.
Algebra Universalis 17 (1983), no. 2, 191-199.
From the Math Review (written by Steve Comer):
The authors …
- 14.6k
7
votes
Is the class of power-associative binars finitely axiomatizable?
The question has been answered, but I will add some remarks
about magma/groupoid/binar. This is in response to some
of the comments on this page:
What you can currently read on the English Wikipedi …
- 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