# 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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### Are there 45 unital magmas with three elements (up to isomorphism)?

How many unital magmas (magma with an identity element) with three elements are there (up to isomorphism)? My approach: List out all of the possible 2x2 multiplication tables for the two non-identity ...
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### Why did I never learn about magmas?

While I’ve never taken an actual abstract algebra course, there are some things I know about the typical curriculum structure: First, define an algebraic structure. Explain groups. Everything else. ...
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### Compilation of Phenomena Modeled by an Operation Table

It seems like there would be utility in a search engine or database through which the user inputs the operation table of a magma (I think that's the right level of algebraic structural generality) to ...
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### Counting the number of points on a curve over a finite field by calculators

I want to count the number of points on a algebraic curve $C:y^2=x^5-x+1$ over $\mathbb{F}_{3^n} (n=2,3,4,...)$ by calculators (Pari/GP, Sage, Magma,...). Can you give me a command that solves the ...
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### Generalization of free magmas for nested structures

Consider a nonempty set $X$. What is the name / concept that gives rise to (the set of) all $X$ labeled planar trees e.g. ...
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### Does the percentage of associative operations on a finite set decrease monotonically towards zero?

In this answer, André Nicolas proves that it is rare for a binary operation on a finite set to be associative, in the following sense: if $A_n$ denotes the number of semigroups that can be defined on ...
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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 ...
1 vote
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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 ...
1 vote
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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{ ...
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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-...
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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, ...
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### if $\cdot$ and $\odot$ are associative operations on $\mathbb{Z}$ when is the sum $(\cdot + \odot)$ associative?

Where $a(\cdot + \odot)b$ is defined as $(a\cdot b) + (a\odot b)$. I know if $\cdot$ and $\odot$ distribute through addition (i.e. $a\cdot(b+c)=a\cdot b+ a\cdot c$) then the sum $(\cdot + \odot)$ is ...
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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 $t$?
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