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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Program to calculate homology of a Koszul complex involving univariate polynomials
Let $R = \mathbb{Z}[x_1,...,x_6]$ be a polynomial ring. Then we may form the Koszul complex $K(x_1,...,x_6)$ which looks something like:
$$ R \xrightarrow{d_6} R^6 \xrightarrow{d_5} R^{15} \...
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Is there a concept representing "connectedness" in abstract algebra?
Consider an object, call it a web, that consists of a set $S$ equipped with a binary operation obeying these axioms:
$$
\forall\ a,b \in S\ \exists\ c \in S :a\ \bullet\ b=c
$$
$$
\forall\ a,b \in S\ \...
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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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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 \...
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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=...
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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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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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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: $...
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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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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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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)...
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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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Good book for self-study of Magmas/Semigroups/etc.?
I'm currently an undergrad in my second semester of Abstract Algebra. We've covered groups, rings, fields, all that fun stuff. I'm working with Shahriari's "Algebra in Action" as well as ...
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What is a monoid in simple terms?
I encountered the term "monoid" but I didn't really understand what is it useful for or what's it about.
If I understand correctly a "monoid" is something defined in the context of ...
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What is the name for a magma which is neither a quasigroup nor a semigroup yet has both an identity and inverses?
Is there a name which is more specific than `unital magma' for a magma whose only requirements are that it should have both an identity and (L/R symmetric) inverses for all elements?
The following ...
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Suspicious diagrams on wiki about group-like structures
It seems to me that the diagrams on wiki about group-like structures are not quite right. For example, the following
https://en.wikipedia.org/wiki/Monoid#/media/File:Algebraic_structures_-...
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Recursive definition of products
Let $(M,*)$ be a Magma. How can one recursively define products such as $(a_1*a_2)*(a_3*(a_4*a_5))$ and so on ?
The basic idea is i think that we have something like : $P^1(a_1)=a_1$ and $P^n(a_1,....,...
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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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