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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57 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-...
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1answer
69 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 ...
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2answers
82 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 ...
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1answer
68 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" $\phi : A \to M$ ...
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3answers
128 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{...
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1answer
47 views

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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31 views

Frequency of items in a list in MAGMA

I'm working in MAGMA and have ended up with a sequence with repeated entries. e.g. something somewhat (although much larger in my application) like this: ...
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0answers
52 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 ...
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31 views

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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77 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 ...
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1answer
77 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)...
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156 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-...
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1answer
177 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, ...
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1answer
64 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 ...
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1answer
109 views

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 $t$?
5
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107 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 ...
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1answer
167 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 ...
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2answers
72 views

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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157 views

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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40 views

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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2answers
125 views

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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1answer
150 views

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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1answer
129 views

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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1answer
105 views

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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1answer
84 views

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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1answer
71 views

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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199 views

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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1answer
60 views

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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1answer
37 views

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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1answer
60 views

Power associative magma

I’m looking for a magma with specific properties: Requirements: 1.Power Associative(of course, I want it to not be alternative or similar). 2.Invertibility and identity element. Preferences(In order ...
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1answer
60 views

"Equivalence relation compatible w/magma law" in Bourbaki's Algebra I

I am using the edition of Bourbaki's "Algebra I" published/printed by Springer in 1989. On p. 11 Bourbaki defines the compatibility between a magma law ⊤ and an equivalence relation R on the ...
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1answer
43 views

Notation and terminology for free algebras with one binary operation?

Introduction To Question Context: Universal Algebra I Definition: A $\mathtt{S}$-algebra is an algebra $\langle A, succ, \bullet \rangle, $ with one unary operation and no identities. Let $\mathsf{S}(...
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1answer
56 views

Is totally ordered magma infinite?

It is well known that any non-trivial totally ordered group is infinite. Is it true that any totally ordered magma with more than one element is infinite too? My attempt to prove the statement: Let'...
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1answer
45 views

Principal ideal of a non-associative magma

The definitions of a left, right, and two-sided ideal of an algebra do not involve associativity (R.D. Schafer "An Introduction To Nonassociative Algebras"). The same we can say about the definitions ...
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0answers
29 views

How to construct a free magma without using identifications/disjoint union?

Let $S$ be a set. We define a sequence of sets $(S_n)_{n\in\mathbb{N}^*}$ recursively as follows: $S_1=S$ and for $n\ge2$ $$S_n=\bigcup_{k=1}^{n-1}\{k\}\times\big(S_k\times S_{n-k}\big).$$ Let $...
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37 views

Every submagma of a free magma is free

Let $X$ be a set. Let $M_X$ be the free magma constructed on $X$. Suppose $N\subset M_X$ is a submagma of $M_X$: i.e. $NN\subset N$. Let $u:(N-NN)\rightarrow N$ be the canonical injection. We know ...
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39 views

Each magma $M$ is associated with monoids $\mathcal{L}(M)$ and $\mathcal{R}(M)$. What are these called, and have they been studied?

Let $X$ denote a magma. Then $\mathrm{List}(X)$ is a monoid equipped with both a left and a right action on $X$, where the actions are defined in the obvious way. To illustrate these actions, suppose ...
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1answer
38 views

Generated submagma of a free magma

Let $X$ be a set and $S\subset X$. Let $M(X)$ denote the free magma constructed on $X$ and $i:S\hookrightarrow X $ be the canonical injection of $S$ into $X$. We know that there exists a unique ...
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1answer
120 views

What is the identity element of 4

If $*$ is a binary operation taking the greater of two distinct numbers, construct a table for the operation on the set $S=\{1,2,3,4,5\}$. What is the identity element of 4? Is the operation ...
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1answer
113 views

Inverse element in a magma [closed]

Given $(S,*)$ a magma and an identity element $e$. The inverse of $x\in S$ is $y$ such that $x*y=e=y*x$. Is it correct to say that if $x$ is the inverse of $y$ then $y$ is the inverse of $x$?
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1answer
82 views

What's the name for a datatype which has every property of a group except commutativity instead of associativity?

I have a set X and an action Y where: Closure: for every element x1 and x2 in X, x1Yx2 is also in X Identity: there is an element e in X where for all x in X, eYx = xYe = e There is at least one ...
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1answer
72 views

Magma $(\mathbb R,*)$ with a binary operation $\;a*b=a+b-2a^2b^2$

Let $(\mathbb R, *)$ be a magma with a binary operation: $$a*b=a+b-2a^2b^2$$ Prove $(a)$ the binary operation is commutative, but not associative, $(b)$ $0$ is a neutral element for that ...
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7answers
96 views

Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these?

Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these? I tried to solve it by assuming $ a,b,c \in G $ such that $a*(b*c)=(a*b)*c$. Then$$\frac{a+\frac{b+c}{2}}{2} = \frac{...
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3answers
362 views

What is difference between idempotent magma and unital magma?

I don't understand well in what way idempotent element is wired to identity element in a magma context. idempotent: $x \cdot x = x$ identity element: $1 \cdot x = x = x \cdot 1$ For example ...
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2answers
37 views

Valid reason to prove the dis-associativity of ($\mathbb R, -)$.

Is this a valid proof? Let $a,b,c$ $\in$ $(\mathbb R, -)$. We want to prove that $(\mathbb R, -)$ does not have an associative property. Assume that it is associative such that: $a-(b-c) = (a-b)-c$. $...
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1answer
89 views

All magmas of order n (specifically 3)

I am considering the collection of all magmas (sets with binary operations) of order 3. Since we just need a binary operation and no other properties, it makes sense to define a magma in terms of all ...
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0answers
40 views

operation on set proof

Consider the operation ⊥ defined by placing, for every $x,y\in Z$ $x⊥y=x+|y|$, Check Associativity and Commutativity. Is there a Identity element in $Z$? My proof: Associativity $x⊥(y⊥z)=(x⊥y)⊥z$ $x⊥(...
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1answer
41 views

Is the Binary Operation ever Invertible in a Semigroup?

A semigroup is a set $S$ together with an associative binary operation $m:S\times S\rightarrow S$. In any kind of semigroup I can think of (group, ring, field, etc), this binary operation $m$ is not ...
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2answers
273 views

Prove that * is commutative and associative

Assume that $*$ is an operation on $S$ with identity element $e$ and that $x*(y*z)=(x*z)*y$ for all $x, y, z$ in $S$. prove that $*$ is commutative and associative Ok, I know that in order for it ...
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2answers
116 views

Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions?

Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions? I have some reasonable pre/post graduate Math skills but no access to Magma etc. I suspect there are none other than ...