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 ...
Joel K's user avatar
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5 answers
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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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Nomenclature for a unital magma together with a monoid

Is there some established name/nomenclature for structures $\mathfrak{A} = (A,\, {\oplus},\, {\odot})$, where $(A,\, {\oplus})$ forms a (commutative) unital magma (in particular not associative!), $(...
blk's user avatar
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1 answer
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Maximal Extension Chain of Halfgroupoids

A book I am reading gives the following definitions: A collection $\{L_i:i=0,1,2,...\}$ of halfgroupoids $L_i$ is called an extension chain if $L_{i+1}$ is an extension of $L_i$ for each $i$. If $G$ ...
shea's user avatar
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Closest Equivalent to Cayley Graphs for Partial Groupoids?

[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.] This question may be nonsensical, given that the duality ...
shea's user avatar
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1 answer
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Does 2nd power idempotency imply all nth powers idempotency?

Suppose $(M,*)$ is a magma, that is, just a set with a binary operation with no conditions imposed, and let $s$ be an element of $M$. Also, let $n$ be an integer greater than or equal to $2$. An $n$-...
user107952's user avatar
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Does there exist a magma where every element has a left cube root but not every element has a right cube root?

Let $(M,*)$ be a magma. $x$ is said to be a left cube root of $y$ if $(x*x)*x=y$. $x$ is said to be a right cube root of $y$ if $x*(x*x)=y$. Does there exist a magma where every element has a left ...
user107952's user avatar
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Is there an algorithm for finding every isolated singular point on an algebraic variety, or a programming language that implements this?

Suppose one wishes to test if a given algebraic surface f(x,y,z,w) = 0 in projective 3 space has singular points, that is df/dx = df/dy = df/dz = 0, and one also wishes to calculate these singular ...
drfpslegend's user avatar
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1 answer
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Term for a Set Equipped With a Binary Operation Which Contains Inverses

Let $A$ be a set and let $\circ:A\times A\rightarrow B,$ $A\subseteq B$ be a binary operation ($A$ is not necessarily closed under $\circ$). If there exists some unique $e\in A$ such that $e\circ a=a\...
Miles Gould's user avatar
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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 ...
user10478's user avatar
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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 ...
user682141's user avatar
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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. ...
Olivier Bégassat's user avatar
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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 ...
Joe's user avatar
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2 votes
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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 ...
Xuesong Si's user avatar
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1 answer
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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 ...
user107952's user avatar
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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. ...
user107952's user avatar
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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 ...
Mithrandir's user avatar
3 votes
1 answer
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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 ...
mathlander's user avatar
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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 ...
scsnm's user avatar
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1 answer
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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-...
user107952's user avatar
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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 ...
HalfTea's user avatar
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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) ...
misanek123's user avatar
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1 answer
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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} \...
Dylan's user avatar
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1 answer
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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\ \...
Eden Laika's user avatar
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1 answer
105 views

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?
user107952's user avatar
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2 votes
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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 \...
BatMath's user avatar
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1 answer
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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=...
user107952's user avatar
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1 vote
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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 $(S, \odot)$ ...
Prime Mover's user avatar
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0 votes
1 answer
66 views

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 ...
user107952's user avatar
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0 votes
0 answers
56 views

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,...
user107952's user avatar
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1 vote
1 answer
96 views

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 ...
Tyrrell McAllister's user avatar
1 vote
0 answers
56 views

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{ ...
Tyrrell McAllister's user avatar
1 vote
1 answer
93 views

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: $...
Max Li's user avatar
  • 63
2 votes
2 answers
90 views

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-...
Lukas Juhrich's user avatar
4 votes
1 answer
139 views

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 ...
Harry Altman's user avatar
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2 votes
2 answers
251 views

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 ...
Math1's user avatar
  • 23
4 votes
1 answer
185 views

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" $\phi : A \to M$ ...
jskattt797's user avatar
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4 votes
3 answers
372 views

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{...
Math_Day's user avatar
  • 1,197
0 votes
1 answer
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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 ...
Mr. J's user avatar
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1 vote
1 answer
187 views

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 ...
Alex C's user avatar
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0 answers
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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. ...
opfromthestart's user avatar
3 votes
2 answers
158 views

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 ...
Jean Du Plessis's user avatar
5 votes
1 answer
86 views

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)...
Noah Schweber's user avatar
5 votes
3 answers
197 views

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-...
Noah Schweber's user avatar
5 votes
1 answer
216 views

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, ...
user107952's user avatar
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1 vote
1 answer
106 views

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 ...
Mars's user avatar
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4 votes
1 answer
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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 $t$?
user107952's user avatar
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5 votes
2 answers
443 views

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 ...
BabylonianTriple's user avatar
1 vote
1 answer
2k views

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 ...
Jim's user avatar
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