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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.
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
1
vote
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
6
votes
Accepted
A ‘canonical’ bounded lattice with proper de Morgan negation?
Let $\phi=(p\vee(q\wedge r))\wedge (r\vee(p\wedge q))$ and let $\chi = (p\wedge (r\vee (p\wedge q)))\vee (r\wedge (p\vee (q\wedge r)))$. For your De Morgan lattice ${\bf M}$ we have $\phi\models_{\bf …
- 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
Finite-join antichains in lattices
I will add a few comments to Bjørn Kjos-Hanssen's answer.
Does property (A) have a name in the literature? Is it a studied notion?
The property is a kind of independence property. A set-th …
- 14.6k
2
votes
Accepted
Injections without fixed-points and the Axiom of Choice
It is shown in
Tachtsis, E.
On the existence of permutations of infinite sets without fixed points in set theory without choice.
Acta Math. Hungar. 157 (2019), no. 2, 281-300.
that ZF+(every infinite …
- 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
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
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
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
6
votes
Accepted
Algebras admitting quantifier elimination
The finite groups with quantifier elimination are classified in
Cherlin, Gregory; Felgner, Ulrich, Homogeneous finite groups.
J. London Math. Soc. (2) 62 (2000), no. 3, 784–794.
- 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
4
votes
Accepted
Do almost-point-transitive algebras generate almost-point-transitive varieties?
Let me distinguish between clone and polynomial clone.
The former is the smallest composition-closed
collection of operations on $A$
containing the primitive operations of $\mathbb A$ and the projecti …
- 14.6k