All Questions
Tagged with magma universal-algebra
18
questions
1
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46
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Is there a model of this equational theory which is not power-associative?
This is a follow up to my previous question, here: Two questions regarding equational axiomatizations of power-associative magmas.. As before, let $t$ be a term, in the sense of universal algebra. I ...
- 21.5k
1
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53
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Two questions regarding equational axiomatizations of power-associative magmas.
A power-associative magma is a magma $(M;*)$ where the submagma generated by any single $x$ in $M$, is associative. I have two questions regarding power-associative magmas. First, some terminology. ...
- 21.5k
0
votes
1
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40
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An equational basis for the variety generated by the following class of magmas.
Let $(M;*)$ be a magma, and define a binary relation $R$ on $M$ by saying that $aRb$ iff there exists a $c$ in $M$ such that $a*c=b$. I call $R$ the left-divisor relation associated with $(M;*)$. I ...
- 21.5k
2
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1
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107
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Is there a magma with this property?
Does there exist a magma $(S,*)$ such that the only quasi-identities that $*$ satisfies are the trivial ones? And if so, can someone give me an explicit example of such a magma?
- 21.5k
2
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2
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98
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Can we derive associativity of symmetric difference from its simpler properties?
The symmetric difference $Δ: 𝒫(X)×𝒫(X) →𝒫(X)$ has a few obvious properties:
$∅$ acts as the neutral element, i.e. $SΔ∅ = S$
It is commutative
Every element is its own inverse.
The (imo) only non-...
- 3,415
5
votes
1
answer
160
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Do the Moufang identities *themselves* imply diassociativity / Moufang's theorem / Artin's theorem?
A Moufang loop is a loop satisfying the Moufang identities. Famously, these are diassociative -- the subloop generated by any two elements is associative (is a group) -- and more generally, they ...
- 4,742
5
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87
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Rings with primal term reducts
This question is a follow-up to this one.
Say that a term reduct of a ring $\mathcal{R}=(R; +,\times,0,1)$ is a magma $\mathcal{M}$ whose domain is $R$ and whose magma operation is $(x,y)\mapsto t(x,y)...
- 249k
5
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3
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197
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Finite magmas representing all unary functions by terms
Say that a magma $\mathcal{M}=(M;*)$ is unary-rich iff for every function $f:M\rightarrow M$ there is a (one-variable, parameter-free) term $t_f$ such that $t_f^\mathcal{M}=f$. For example:
The one-...
- 249k
5
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1
answer
220
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The ratio of finitely based magmas to all magmas
Let $n$ be a positive integer. By $S_n$, I denote the set of positive integers from $1$ to $n$. By $F_n$, I denote the cardinality of the set of magmas on $S_n$ which are finitely based, that is, ...
- 21.5k
4
votes
1
answer
130
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Finite magma where the only equations are of the form "$t=t$"?
Does there exist a finite set $S$ with a single binary operation $*$, where the only equational identities that hold are of the form $t=t$ for some term $t$?
- 21.5k
1
vote
1
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57
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Notation and terminology for free algebras with one binary operation?
Introduction To Question
Context: Universal Algebra
I
Definition: A $\mathtt{S}$-algebra is an algebra $\langle A, succ, \bullet \rangle, $ with one unary operation and no identities.
Let $\mathsf{S}(...
- 1,642
2
votes
1
answer
456
views
In the coproduct of monoids $A\amalg B$, suppose two words have the same suffix in $B$. Can it be cancelled?
Let $A,B$ be monoids and $A\amalg B$ their product in the category of monoids, comprised of reduced words. Previously I have asked about the canonical arrow $A\amalg B\to A\times B$ given by e.g $$...
- 14k
4
votes
1
answer
387
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Elegant approach to coproducts of monoids and magmas - does everything work without units?
From this answer by Martin Brandenburg I learned an elegant way of constructing and describing elements of coproducts of monoids. There he writes this construction shows the existence of colimits in ...
- 14k
2
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1
answer
127
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Concretely describing the arrow $A\amalg B\to A\times B$ for nonlinear algebraic theories
I'm reading about unital categories to get a better understanding of nonlinear algebraic categories like groups, monoids, semigroups, magmas etc.
A unital category is a pointed finitely complete ...
- 14k
3
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1
answer
223
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Varieties of groupoids which aren't definitionally equivalent
Here is the exercise from Smirnov's book "Varieties of algebras" (in Russian).
Problem: Let $\mathcal{U}$ be the variety of all groupoids $(A, \cdot)$ and $\mathcal{V}$ be the variety of all ...
- 2,906