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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Conservative idempotent magma - proof attempt
I need help with checking proof about idempotent and conservative magmas.
Let magma be any ordered pair $(M, \odot)$, where $M$ is nonempty set and $\odot$ binary operation on $M$.
Now I need to ...
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Is subtraction on the reals isomorphic to division on the positive reals?
I know that the magma $(\mathbb{R};+)$ of addition on the real numbers is isomorphic to the magma $(\mathbb{R}^+;\times)$ of multiplication on the strictly positive real numbers. I wonder, is it the ...
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Weaker notion of closure for partial magmas
Let $(G,\cdot)$ be a partial magma (a set endowed with a partial binary operation). In principle, for such generic structures it is possible that $\exists g \in G$ such that $\forall h \in G, \, g\...
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map from spin to special orthogonal in Magma [closed]
Let $G:=\operatorname{Spin}(7,5)$. How to construct in Magma the map $G \rightarrow G/Z(G) $ where $Z(G)$ is the center. I get this from Magma:
...
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Practical example of differences between associativity and alternativity (and the in-between Bol loop)? [closed]
Associativity is:
$$(a * b) * c = a * (b * c)$$
Alternativity is:
$$a * (a * b) = (a * a) * b$$
$$(a * b) * b = a * (b * b)$$
Bol loop is:
$${\displaystyle a(b(ac))=(a(ba))c}$$
$${\displaystyle ((ca)b)...
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Smallest possible cardinality of finite set with two non-elementarily equivalent magmas which satisfy the same $\forall$-theory?
This is a follow-up to my previous question, here: Smallest possible cardinality of finite set with two non-elementarily equivalent magmas which satisfy the same quasi-equations?. My question now is, ...
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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.
...
user1286095
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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!),
$(...
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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$ ...
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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 ...
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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$-...
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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 ...
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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 ...
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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\...
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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 ...
- 1,016
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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 ...
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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 ...
- 21.5k
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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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0
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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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1
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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{...
- 1,255
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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 ...
- 1,120
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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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