# 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.)

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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-...
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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 ...
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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 ...
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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" $\phi : A \to M$ ...
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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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### Simplification of a group presentation

Im new to MAGMA and hope somebody will help me with my question. If a group has a presentation with 4 generators, is there a magma code/function that can give me the same group with only three ...
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### Frequency of items in a list in MAGMA

I'm working in MAGMA and have ended up with a sequence with repeated entries. e.g. something somewhat (although much larger in my application) like this: ...
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### Inverse element of a magma

It is accepted that two elements are inverse to each other if their product is equal to the identity element: Inverse element in a magma https://en.wikipedia.org/wiki/Inverse_element The definition ...
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### Isomorphisms of magmas that are subsets of R

Let there be two sets $A, B\subseteq\Bbb{R}$ and let there be two binary operations $*_M$ and $*_N$. Under what circumstances is $(A,*_M)\cong(B,*_N)$? I have found a couple of general working cases. ...
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### How many non-isomorphic algebraic structures (i.e. magmas, monoids, groups etc.) are there with countably infinite order? [closed]

For structures of finite order it seems obvious to me that there are countably infinite in total, by a simple diagonalization argument (starting at all of order 1, then 2 etc.). It is however not ...
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### Defining loops: why is divisibility and identitiy implying invertibility?

Wikipedia contains the following figure (to be found, e.g. here) in order to visualize the relations between several algebraic structures. I highlighted a part that I find especially interesting. It ...
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### When do we have $(x y)^2 = x^2 y^2$?

I just started thinking about algebra so this might be a trivial question. Anyway, Under what conditions do we have $$(x y)^2 = x^2 y^2$$ ? Does it need to be a group ? Or a groupoid ? Or a monoid ? ...
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### Computing the number of conjugacy-classes in $GL_{n}(\mathbb{F}_{p})$ of elementary abelian p-subgroups by GAP and Magma

I'm trying to compute the number of conjugacy-classes of elementary abelian p-subgroups of rank $2$ in $GL_{n}(\mathbb{F}_{p})$ by GAP and Magma. So I consider the following GAP function: ...
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### Is every group isomorphic to the automorphism group of some magma?

I believe that magma isomorphism is defined as $\phi(x*y)=\phi(x)*'\phi(y)$. The automorphism group is the set of bijective isomorphisms from the elements of the magma to itself, under the operation ...
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### What's the preferred term researchers like to use in the theory of magmas/groupoids?

As we know, mathematicians like to avoid the term "groupoid" to refer to a set with binary operation. This term, as we know, originates from the works of Brandt, so called Brandt groupoid. A ...
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### Is there a category of partially defined binary operations?

A magma is a set $Y$ with a binary operation $m:Y \times Y \rightarrow Y.$ A partial magma is the same idea, but where the binary operation $m$ may not be defined on some pairs of elements of $Y.$ My ...
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### If not associative, then what?

Consider a binary operation $*$ acting from a set $X$ to itself. It's useful and standard to work with operations which are associative, such that $(a*b)*c = a*(b*c)$. What about operations which are ...
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### Fill in a partly filled in table such that it makes the magma $(M,*)$ associative, commutative, has an identity element and has no zero-elements.

Below is a partly filled in table for a binary operation ($*$) on the set $M=\{a,b,c,d\}$. I am trying to fill in the rest such that the magma $(M,*)$ becomes associative, commutative, has an identity ...
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### Are all alternative magmas flexible?

A magma is alternative if for all elements $x$ and $y$, we have $(xx)y = x(xy)$ and $y(xx) = (yx)x$. A magma is flexible if for all elements $x$ and $y$ we have $x(yx) = (xy)x$. Both of these are ...
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### Power associative magma

I’m looking for a magma with specific properties: Requirements: 1.Power Associative(of course, I want it to not be alternative or similar). 2.Invertibility and identity element. Preferences(In order ...
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### "Equivalence relation compatible w/magma law" in Bourbaki's Algebra I

I am using the edition of Bourbaki's "Algebra I" published/printed by Springer in 1989. On p. 11 Bourbaki defines the compatibility between a magma law ⊤ and an equivalence relation R on the ...
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### Every submagma of a free magma is free

Let $X$ be a set. Let $M_X$ be the free magma constructed on $X$. Suppose $N\subset M_X$ is a submagma of $M_X$: i.e. $NN\subset N$. Let $u:(N-NN)\rightarrow N$ be the canonical injection. We know ...
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### Each magma $M$ is associated with monoids $\mathcal{L}(M)$ and $\mathcal{R}(M)$. What are these called, and have they been studied?

Let $X$ denote a magma. Then $\mathrm{List}(X)$ is a monoid equipped with both a left and a right action on $X$, where the actions are defined in the obvious way. To illustrate these actions, suppose ...
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### Generated submagma of a free magma

Let $X$ be a set and $S\subset X$. Let $M(X)$ denote the free magma constructed on $X$ and $i:S\hookrightarrow X$ be the canonical injection of $S$ into $X$. We know that there exists a unique ...
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### What is the identity element of 4

If $*$ is a binary operation taking the greater of two distinct numbers, construct a table for the operation on the set $S=\{1,2,3,4,5\}$. What is the identity element of 4? Is the operation ...
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### Inverse element in a magma [closed]

Given $(S,*)$ a magma and an identity element $e$. The inverse of $x\in S$ is $y$ such that $x*y=e=y*x$. Is it correct to say that if $x$ is the inverse of $y$ then $y$ is the inverse of $x$?
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### What's the name for a datatype which has every property of a group except commutativity instead of associativity?

I have a set X and an action Y where: Closure: for every element x1 and x2 in X, x1Yx2 is also in X Identity: there is an element e in X where for all x in X, eYx = xYe = e There is at least one ...
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### Magma $(\mathbb R,*)$ with a binary operation $\;a*b=a+b-2a^2b^2$

Let $(\mathbb R, *)$ be a magma with a binary operation: $$a*b=a+b-2a^2b^2$$ Prove $(a)$ the binary operation is commutative, but not associative, $(b)$ $0$ is a neutral element for that ...
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### Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these?

Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these? I tried to solve it by assuming $a,b,c \in G$ such that $a*(b*c)=(a*b)*c$. Then\frac{a+\frac{b+c}{2}}{2} = \frac{...
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### What is difference between idempotent magma and unital magma?

I don't understand well in what way idempotent element is wired to identity element in a magma context. idempotent: $x \cdot x = x$ identity element: $1 \cdot x = x = x \cdot 1$ For example ...
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### Is the Binary Operation ever Invertible in a Semigroup?

A semigroup is a set $S$ together with an associative binary operation $m:S\times S\rightarrow S$. In any kind of semigroup I can think of (group, ring, field, etc), this binary operation $m$ is not ...
Assume that $*$ is an operation on $S$ with identity element $e$ and that $x*(y*z)=(x*z)*y$ for all $x, y, z$ in $S$. prove that $*$ is commutative and associative Ok, I know that in order for it ...
### Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions?
Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions? I have some reasonable pre/post graduate Math skills but no access to Magma etc. I suspect there are none other than ...