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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24 views

Why to have an inverse element (for example taken from bijection definition) do we need start from an unital magma?

I pictured something like the set of the Real numbers, with “1” removed. Every number in the set has a multiplicative inverse, and the set lacks an identity element. One could start with axioms ...
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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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“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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List of equations in MAGMA

I have some integer $n$, some ambient affine space $\mathbb{A}^n$, and a list $L$ of equations $f_{ij}$ cutting out a variety $X$ in the ambient space. I have problems defining the list $L$ correctly. ...
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How does one find functionally complete sets, or sole sufficient operators, on more than two truth values?

In standard two-valued logic, it is known that either NAND or NOR is sufficient by itself to construct all possible logic gates - all functions 2 × 2 → 2. But I've not been able to find anything about ...
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31 views

Direct decomposition of a magma onto ideals

Let's call the product of submagmas $A \cdot B$ a direct decomposition of a magma $M(\cdot)$ if: $A \cdot B = M$ ($m = a \cdot b$ for any element $m$ of $M$, where $a$ is an element of $A$ and $b$ is ...
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1answer
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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
45 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
32 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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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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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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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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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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Extending a mapping from a set to a morphism from the free magma on the set

Let $X$ be a set. We define a sequence of sets $(M_n(X))_{n\ge 1}$ recursively as follows: $M_1(X)=X$ and for $n\ge2$ $$M_n(X)=\bigcup_{p=1}^{n-1}\big(M_p(X)\times M_{n-p}(X)\big)\times\{p\}.$$ Let $M(...
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Construction of free magma

Let $X$ be a set. Let $M_1(X)=X$, and for every natural number $n\geq2$ let $$M_n(X)=\bigcup_{p=1}^{n-1}\big(M_p(X)\times M_{n-p}\big)\times\{p\}.$$ Then let $$M(X)=\bigcup_{n\geq 1} M_n(X)\times\{n\}...
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1answer
63 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
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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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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
48 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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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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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
36 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
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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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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
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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
111 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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Reference Request: An operation preserving bijection from a magma to a group must be a group isomorphism

Let $M$ be a magma with a binary operation $*_M$ and let $G$ be a group with a binary operation $*_G$. If $f$ is a bijection from $M$ to $G$ preserving the operation, that is, $f(m_1 *_M m_2)=f(m_1)...
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1answer
89 views

Given an algebra structure $(X,*)$ s.t. $(x*y)*y = y*(y*x) = x$ , prove$x*y=y*x$.

Suppose $(X,*)$ is arbitrary algebraic structure such that $\forall x,y\in X$, we have $(x*y)*y = y*(y*x) = x$, prove that $x*y=y*x$. This question seems pretty simple but I tried and I failed.
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Commutative subtraction

It is well known that subtraction is not commutative in general. However, it is commutative in some groups: $\mathbb I$, $\mathbb C_2$, $\mathbb K_4$. I am trying to understand the logic. ...
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45 views

How many magmas exist on $n$-element set

It is clear that we can make $n^{n^2}$ Latin squares (I think that this is no real Latin square, but I don't know how to name it) for $n$-element set, but I have heard that some magmas will be ...
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Elements $A\in GL_4(\mathbb{Z}_2)$ with $A^5=I, A\neq I$ by using GAP.

I need elements $A\in GL_4(\mathbb{Z}_2)$(General linear group of $4\times 4$ matrices over $\mathbb{Z}_2$ ) with $A^5=I, A\neq I.$ By using simple calculation its hard to find such types of elements....
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Have a magma structure when “if the set of integers with respect to subtraction is not a group”? [closed]

I have a 3 answers but nobody return me a mathematical structure/category name when I try to classify "the set of integers with respect to subtraction is not a group" 1) Subtraction of integers (and ...
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2answers
226 views

Magma function for modulo irreducible polynomial

So, I am trying to make a program in Magma which returns the value table of a given function F over a field $GF(2^n)$. To do so I need a irreducible polyomial. For example, I've considered $GF(2^3)$ ...
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Invertibility as Criteria for a Loop

I try to understand the correct criteria for a Loop. I see in Wikipedia https://en.wikipedia.org/wiki/Inverse_element#In_a_unital_magma that “A unital magma in which all elements are invertible is ...
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1answer
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Generating subsets of a finite magma

I am trying to write a program which, given a multiplication table of a finite magma $(G, *)$, should produce at least one (or all possible) generating subset $S$ of minimal cardinality. More ...
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Question related to Magma

I am reading some notes in which I found the following exercise: Suppose $G$ is a magma then $G$ is associative and satisfy cancellation properties. I think this is not true for instance matrix ...
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2answers
190 views

Notions of basis and span in a magma

Suppose that $C$ is a set with closure under the binary operation $+$. $(C,+)$ is therefore a magma. I am trying to figure out if notions of basis, or span make sense in a magma. Spanning set (?) ...
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1answer
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Prove that there is no bijective homomorphism from $\left(\mathbb{Q},\ +\right)$ to $\left(\mathbb{Q_+^*},\ \times \right)$

I need to prove that there does not exist any bijective homomorphism from $\left(\mathbb{Q},\ +\right)$ to $\left(\mathbb{Q_+^*},\ \times \right)$ Here is a way to prove it: Let $f$ be a ...
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Subtraction Magmas

I was looking at a collection of related closed binary operations on sets (magmas): Subtraction on the integers, reals, etc. Set difference Set symmetric difference Saturating subtraction on the ...
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How many different operations can be defined in a finite groupoid with a given property?

Set $B=\left\{ 1, 2, ... 18 \right\}$ is given. How many different operations $*$ can be defined so that $(B,*)$ is a groupoid with a property that $|\left\{i|i*(19-i) \neq i ∧ i*(19-i) \neq (19-i)\...
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7answers
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What is an example of a groupoid which is not a semigroup?

I know that groupoid refers to an algebraic structure with a binary operation. The only necessary condition is closure. However, I couldn't find any easy-to-understand example of a groupoid which is ...
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1answer
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Uniqueness of two side zeroes of binary operation

I came across the following fact in group theory: Two-sided identity of binary operation is unique. Does the similar statement for two sided zero also holds? : Two-sided zero of binary ...
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1answer
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Is there a name for an algebraic structure with only “addition” and “truncated subtraction”?

Given a set $S$ with An associative binary "addition" operation $+$ with corresponding neutral element $0$ such that $(S, +)$ is a monoid A non-associative binary "truncated subtraction" operation $-$...
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3answers
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Terminology: Semigroups, only their “binary operations” aren't closed.

Motivation: Consider $\mathcal{X}=(X, +)$, where $X=\{-1, 0, 1\}$ and $+$ is standard addition. Then $\mathcal{X}$ is associative (where defined) but not closed. NB: There is an identity element in $X$...
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Structures with $x*(y*z) = y*(x*z)$

In reading http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=182143561104878BDABB72258DA254D0?doi=10.1.1.18.2521&rep=rep1&type=pdf , they mentioned an interesting relation -- they had a ...
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Is “(a * a') is cancellative” + “M has an identity” the same as “a has an inverse”

Given a magma $(M, \ast)$, $(a \ast a')$ is cancellative, iff $$\forall b,c \in M. b \ast (a \ast a') = c \ast (a \ast a')\Leftrightarrow b = c$$ The magma has an identity, iff: $$\exists e.\forall ...
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1answer
187 views

Group Theory For an algebraic system, how to prove the following?

I'm trying to prove the below equation (From Elements of discrete mathematics, second edition by C. L. Liu Question 11.13) Let $(A, +)$ be an algebraic system such that for all $a, b$ in $A$ we ...
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“Compact” formula for counting all products of $x_1,\ldots,x_n$(in that order)?

So here is the problem (I.1.3 in Grillet's Abstract Algebra) Let $X$ be a set with a binary operation $\cdot:X \times X \to X,$ where $\cdot(x,y):= xy, \text{ for all } x,y\in X$. A product $x\in X $...
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1answer
330 views

how to use magma to find the rank of the elliptic curve

I am trying to find as much as possible of elliptic curves in Magma. What is the code for finding the rank of elliptic curve in Magma? I want to write a for loop for the curve $y^2=y^3+ax^2+bx+c$ for (...
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1answer
77 views

Sufficient condition for a magma to be a topological magma

Let $(B,\ast)$ be a magma (that is, $\ast:B\times B\to B$ is a binary operation on $B$), and let $\tau$ be a topology on $B$. If $X$ is any set and we define $\tilde\ast$ in the set $B^X$ of functions ...