Questions tagged [magma]
A magma is a set together with a binary operation on this set. (For questions about the computer algebra system named Magma, use the [magma-cas] tag instead.)
196
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Is there a concept representing "connectedness" in abstract algebra?
Consider an object, call it a web, that consists of a set $S$ equipped with a binary operation obeying these axioms:
$$
\forall\ a,b \in S\ \exists\ c \in S :a\ \bullet\ b=c
$$
$$
\forall\ a,b \in S\ \...
- 31
2
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1
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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
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How to show that a compact semigroup for which the cancellation law holds is a compact group
Here is my problem:
Set $G$ a compact semigroup (that is a Hausdorff compact space endowed with an associative continuous binary operation). Assume that the cancellation law holds i.e. for any $g,h,k \...
- 166
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1
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Does "in-between" multiplication preserve equality?
In a magma $(S;*)$, multiplication on the left and the right preserves equality. That is, if $a=b$, then $c*a=c*b$ and $a*c=b*c$. But what about "in-between" multiplication? That is, if $a*c=...
- 21.5k
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1
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Given that $f$ and $g$ are homomorphisms, the implication that $f \odot g$ is also a homomorphism implies $(S, \odot)$ is entropic - why?
Context: Seth Warner's "Modern Algebra" (1965), exercise $13.13$. Ongoing self-study.
Let $(S, \odot)$ and $(T, \otimes)$ be closed algebraic structures with one operation. Let $(S, \odot)$ ...
- 5,057
0
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1
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67
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Is there a magma with the following property?
Does there exist an infinite magma with the following property: Let $n$ be a positive integer greater than or equal to $2$. For all $x_1,...,x_n$, if $x_1,...x_n$ are all distinct, then all products ...
- 21.5k
0
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0
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62
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What is the formal definition of a Cayley table?
What is the formal definition of a Cayley table? I am not interested merely in Cayley tables for groups, I am interested in general Cayley tables for non-empty finite magmas. Also, another question is,...
- 21.5k
1
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1
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99
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Term for a semigroup with left identities and left inverses?
Is there a term for a semigroup $(M, *)$ that has at least one
left identity and left inverses in the "weak" sense that, for all
$a \in M$, there exists a $b \in M$ such that $b*a$ is a left
...
1
vote
0
answers
57
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Term for a magma with a left identity?
Is there a term for a magma $(M,*)$ that contains at least one left identity element, but not necessarily a right identity element? I'm looking for a term that requires only
$$\exists e \in M \text{ ...
1
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1
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95
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Algebraic structure for subtraction limited by 0 from below.
Let's assume an algebraic structure with elements from non-negative real numbers with the operation $x - y := max(x - y, 0)$.
It fails at least 2 out of 3 group definition properties:
Associativity: $...
- 63
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 $Δ: 𝒫(X)×���(X) →𝒫(X)$ 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
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1
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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
2
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2
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Faithful permutation representation
excuse me if my question is trivial.
I’m trying to use magma to construct faithful permutation representations of a certain group using the group action that lets the group G acts by the left ...
- 23
4
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1
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Surjective homomorphism into a magma confers all the algebraic properties of the domain
Let $A$ be your favorite algebraic object (group, abelian group, rng, ring, commutative ring, field, module, vector space). Let $M$ be a magma. The image of a "homomorphism" $\phi : A \to M$ ...
- 1,751
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A commutative but not necessarily associative operation
Let $S$ be a set and let $*$ be a binary operation on $S$ satisfying the laws
\begin{align}
x*(x*y) &= y \quad \text{for all } x, y \text{ in } S,\\
(y*x)*x &= y \quad \text{...
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