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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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Solving a system of linear equations of boolean values with magma [closed]

Hello I´m new to magma and this site. My question is the following: I have a system of boolean variable in the following form which I get from a text file: $x_4+x_2+x_0$ $x_3+x_2+x_0+1$ $x_2+x_0+1$ $...
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1answer
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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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1answer
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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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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
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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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WedderburnDecomposition of $FG/J(FG).$

WedderburnDecomposition(GroupRing(GF(p),G)) gives complete wedderburn decomposition for semisimple group algebra $FG$ for finitie group $G$ and finitie field $F$. But if $FG$ is not semisimple then ...
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0answers
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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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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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1answer
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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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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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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 $...
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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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1answer
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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
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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
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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
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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 ...
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1answer
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Does the existence of only right neutral mean that there can only exist right inverse?

Let $(G,\circ $) be groupoid. If there exists only right neutral, does that mean that there can exist only right inverse? To put it in another way, does the existence of only right neutral mean that ...
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1answer
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Idempotent that isn't any of these

Let $A=\left\{x,y,z\right\}$ and $M=\left\{g\mid g:A\rightarrow A\text{ is a function}\right\}$. Is there an element in $M$ that is idempotent but not right absorbing, left absorbing, a right ...
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1answer
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Showing properties of a set of functions

Let $A$ be a set of at least $2$ elements and $M=\left\{q\mid q:A\rightarrow A\text{ is a function}\right\}$. $($a$)$ $f\in M$ is left absorbing if and only if $f$ is a constant function. $($b$)$ $M$...
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1answer
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Name for the property of “being a Cartesian product of arbitrary sets”

Suppose I have a set $S$ of pairs (or, in general, tuples, or objects with two or more parts/attributes/projections). This set may have the property that $$(a, b) \in S \wedge (a', b') \in S \...
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1answer
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What kind of group-like object is this?

Define a magma $\langle G,*\rangle$ with $\forall a,b \in G,\quad a*b \in G $ $\exists 1(a*1=a\;\text{ and }\;1*a=1)$ $\forall a \in G, \exists b (a*b=1)$ $\exists c \forall a \in G, \quad(a*c=1)$...
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Are all additionlike operations on $\mathbb{R}$ of this form?

Suppose we're given real numbers $x,x',y,y' \in \mathbb{R}$ such that $$|x-x'| \leq a, \qquad |y-y'| \leq b.$$ Then $$|(x+y)-(x'+y')| = |(x-x')+(y-y')| \leq |x-x'| + |y-y'| \leq a+b$$ This tells us ...
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1answer
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A right group is the direct product of a group and a right zero semigroup.

This is (most of) Exercise 2.6.6 of Howie's "Fundamentals of Semigroup Theory". The first part is here. The Details: Let $S$ be a semigroup. Definition 1: We call $S$ right simple if $\mathcal R=...
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1answer
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The direct product $G\times E$ of a group $G$ with a right zero semigroup $E$ is a right group.

This is part of Exercise 2.6.6 of Howie's "Fundamentals of Semigroup Theory". I apologise in advance if this is a duplicate. The Details: Let $S$ be a semigroup. Definition 1: We call $S$ right ...
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1answer
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Deducing a lattice is modular.

This is (the second part of) Exercise 2.6.4 of Howie's "Fundamentals of Semigroup Theory". The first part is here. The Details. Definition 1: A lattice $(L, \le, \wedge, \vee)$ is modular if, for ...
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1answer
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Exercise 2.6.1 of Howie's “Fundamentals of Semigroup Theory”.

This is Exercise 2.6.1 of Howie's "Fundamentals of Semigroup Theory". Definition 1: A semigroup $S$ is cancellative if for all $a, b, c$ in $S$, we have both $ca=cb\implies a=b$ and $ac=bc\implies ...
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Does commutativity of $+$ on $\mathbb{N}$ imply the associativity of $+$ on $\mathbb{N}$?

Question: Consider the binary operation on the free commutative magma generated by one element. Is this binary operation also associative? Note: I do not mean the free magma (1)(2) generated by one ...
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Why is a “double-cancellative” operation so weird?

Let $A = \{ \{0\}, \{0,1\} \}$. Let $\bar{A}$ be the family of sets generated by the Cartesian product on $A$. This is a magma $(\bar{A}, \times)$ that has what I am calling a "double cancellative" ...
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1answer
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Operation satysfaying b*(b*a)=a=(a*b)*b for all a and b must be commutative

Let $X$ be a nonempty set and $*$ operation defined on elements of $X$ such that for $a, b$ from $X$ there is $(a*b)*b=b*(b*a)=a$. Prove that operation $*$ is commutative. Exercise is taken from the "...
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Canonical examples of commutative loops

The context of this question is that I'm trying to understand why groups are so fundamental in modern math (a different question that has already been asked elsewhere). So I was wondering: what if we ...
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Is there a term for a magma with identity (only)?

If we start from magmas and consider: associativity, identity, invertibility (divisibility). We will theoretically get $2^3=8$ structures by regarding whether such structure possess these properties. ...
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Build abelian group containing a set $K$ under an associative, commutative operation $*$ with an identity but the inverses are not always in $K$.

We are given a set K of elements and an operation * . For every element in the set, there exists an inverse element (not necessarily in the set). There are three (additional) rules: 1) K contains e (...
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1answer
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Free magmas and binary trees

The free magma on a set $S$ can be constructed by defining $S_0 = S$, $S_{n+1} := \coprod_{p+q=n} S_p \times S_q$ and then endowing $\coprod_{n \geq 0} S_n$ with the evident magma structure (Bourbaki, ...
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1answer
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In the coproduct of monoids $A\amalg B$, suppose two words have the same suffix in $B$. Can it be cancelled?

Let $A,B$ be monoids and $A\amalg B$ their product in the category of monoids, comprised of reduced words. Previously I have asked about the canonical arrow $A\amalg B\to A\times B$ given by e.g $$...
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1answer
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Elegant approach to coproducts of monoids and magmas - does everything work without units?

From this answer by Martin Brandenburg I learned an elegant way of constructing and describing elements of coproducts of monoids. There he writes this construction shows the existence of colimits in ...
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1answer
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Concretely describing the arrow $A\amalg B\to A\times B$ for nonlinear algebraic theories

I'm reading about unital categories to get a better understanding of nonlinear algebraic categories like groups, monoids, semigroups, magmas etc. A unital category is a pointed finitely complete ...
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Iterating over automorphism group in Magma

I am trying to iterate over the automorphism group of a group but Magma keeps telling me that "Iteration is not possible over this object." Does anyone know a workaround for this? This is what I ...
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Associanize a magma

This is a thing I have been thinking on and gotten a bit frustrated so I share my thoughts here in hope for clarification. Let $M$ be a magma, that is a set with an underlying binary operation which ...
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2answers
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Show that for $(A,*)$ algebraic system $a*(a*b) = a*b$

Suppose we are given that for algebraic system $(A,*)$ that $$(a*b)*a = a $$ and $$(a*b)*b = (b*a)*a$$ then how can I show that $$a*(a*b) = a*b$$ such that $a,b$ belongs to $A$? I know that $(a*b)$...
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2answers
348 views

Is a monoid a magma?

According to this wikipedia page, a monoid is defined as an object that contains An associative binary operation An identity element There is no mention of the object necessarily containing a set. ...
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Is power-associativity an equational property?

A magma $M$ is said to be power-associative if the subalgebra generated by any element is associative. This can be written simply as $x^mx^n=x^{m+n}$ for all $m,n$ positive integers and $x\in M$, ...