# 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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### How to define the non-commutative ring $\mathbb{F}_{4}+e\mathbb{F}_{4}$, $e^2=1$, $ae=ea^2$ in MAGMA(Computational Algebra System)?

I'm trying to learn to use MAGMA(Computational Algebra System) for research in coding theory over non-commutative rings, but it's been slow going. I feel like it's hard to find anything in the ...
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### Is there a finite equational basis for the variety which is the join of the commutative and associative varieties?

I asked this question a while ago, but nobody has really answered it for me. Perhaps it is an open problem. Here is the question: Consider the lattice of equational theories of a single binary ...
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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 ...
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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. ...
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### Eckmann–Hilton Argument and magma homomorphisms

The Eckmann-Hilton result is as follows: Let $X$ be a set equipped with two binary operations $\circ$ and $\otimes$, and suppose $\circ$ and $\otimes$ are both unital, meaning there are identity ...
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### Which axiom can almost determine the magma with one element?

The axiom $((a * b) * c) * (a * ((a * c) * a)) = c$ uniquely determines Boolean algebra, an example of a single axiom giving a magma an "interesting" structure. What is the fewest number of ...
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### intersection of point stabilisers is trivial

Let $G=\operatorname{GL}_{n}(2)$. Let $v_{i}$ be the basis elements of the natural module of $G$. I observed by computing with Magma that the intersection of all Stabiliser($G, v_{i}$) is trivial for ...
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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 ...
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### An example of a binary operation which is neither idempotent nor has a right identity, but has a reflexive "left-divisor" relation.

Let $A$ be a set with a binary operation $*$. I define a binary relation $R$ on $A$ by defining $xRy$ to hold if there exists a $z$ in $A$ such that $x*z=y$, and in that case I call $x$ a "left-...
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### Is class number the always the degree of [Hilbert class field of discriminant $D:K=\mathbb{Q}(\sqrt{d})]$

I was going through https://services.math.duke.edu/~schoen/discriminants.html where the minimal polynomial whose quotient over $K=\mathbb{Q}(\sqrt{d})$ is equal to the Hilbert class field for ...
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### non-commutative algebraic structure with 16 elements, need help categorizing it and finding a representation

We have an abstract algebraic structure with the following multiplication table, has anyone seen this structure before and can anyone give it a proper name and a simple (possibly matrix) ...
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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?
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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 $(S, \odot)$ ...
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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 ...
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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,...
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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 ...
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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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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 ...