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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Just How Strong is Associativity?

A friend of mine is using a lot of Non-associative Algebra for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like "brackets ...
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Number of different magmas up to isomorphism

Let $(M,\circ)$ be a magma over a finite set of order $n$. I tried to count all the possible magmas up to isomorphism, but I just can't get it right. My naive approach was to count all the possible ...
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Functions $h$ such that $h(x*x') = f(x) * g(x').$

Definition 0. Call a magma $X$ surjective iff the distinguished binary operation of $X$ induces a surjective function $X \times X \rightarrow X$. Now for the main idea: Definition 1. Let: ...
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Existence of right and left identity in minimalistic algebraic structure

Let $(A,\cdot)$ be some algebraic structure in which there exists elements $e_r,e_l$ such that $$e_l\cdot x = x, \forall x\in A$$ $$x\cdot e_r = x, \forall x\in A$$ By definition, if $(A,\cdot)$ is ...
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Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
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Isomorphism between two magmas with one.

Do we have a method to find one (or all) isomorphism between two given magmas with one using GAP? Edit If we have Loop or Latin square (with one) instead of Magma then do we have the method?
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GAP code to get Multiplication Table.

I have a finite set $S=\{0,1,2,\ldots,n-1\}$ and binary operation $\star$ on $S$ defined by $$x\star y= \left\{ \begin{array}{l l l} \frac{3(x+y)}{2} ~~\text{modulo} ~~n& \qquad \mbox{if $x$ ...
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Sets as extremely trivial groups

A group is a structure defined upon an underlying set which is endowed with a single binary operator that has some rules attached to it. I was wondering whether one could describe a set itself as ...
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Varieties of groupoids which aren't definitionally equivalent

Here is the exercise from Smirnov's book "Varieties of algebras" (in Russian). Problem: Let $\mathcal{U}$ be the variety of all groupoids $(A, \cdot)$ and $\mathcal{V}$ be the variety of all ...
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Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
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Is there a term for this property of magmas?

There exists an element of the magma c such that for all x: $ x*x=c $ The consequence of this is that the elements on the diagonal of the Cayley table are all the same, e.a: $ * = \begin{bmatrix} 1 ...
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Efficient algorithm for calculating the tetration of two numbers mod n?

I'm trying to study the algebraic properties of the magma created by defining the binary operation $x*y$ to be: $ x*y = (x \uparrow y) \bmod n $ where $ \uparrow $ is the symbol for tetration. ...
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1answer
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Is there a term for an algebraic structure with two binary operators that are closed under a set?

For example, let's say we're using the operators +, and *, and the set {0,1,2} The Cayley tables look like this: ...
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3answers
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Proof of closure and identity element in Abelian Group

For real numbers $x > 1$, which forms the set $G$, it is given that the operation on $a,b$, being $a\ast b$, results in $ab - a - b + 2$ (where $ab$ is the ordinary multiplication of $a$ and $b$). ...
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Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$

When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that $$a*(b*c)=(a\cdot b)*c$$ If $*$ is associative then $\cdot=*$ even if I'm not sure ...
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$\beta_a(n)=(a_1*\cdots(a_n*b))\setminus_* b$ and Iterations in right divisible magmas e representability by left translations.)

Let's consider the magma $(G,*)$ with infinite elements. Now I define $\operatorname{left}(G)$ the set of all the left translations $$\operatorname{left}(G):\{L_a:a \in G ,L_a(b)=a*b\}$$ And ...
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Can you give me some concrete examples of magmas?

I've seen the following: I've learned a bit about groups and I could give examples of groups, but when reading the given table, I couldn't imagine of what a magma would be. It has no ...
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Some questions on interdependence of some properties of abstract magma

Does there exist a magma $(S,\cdot)$ such that for every $y\in S, \exists y'\in S$ such that $x\cdot(y\cdot y')=x, \forall x,y\in S$, but there exist $x_1, x_2, x_3\in S$ such that ...
4
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1answer
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Special Element in a Magma

Say I have a Magma $(M,\times)$, on which there exists an element $k\in M$ such that $k\times x = x\times k = k, \forall x\in M$. $k$ is then called an absorbing element (or zero element). My ...
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Another question on certain magma structures

Let $S$ be a non-empty set and $*$ be a binary operation on $S$ i.e. let $(S,*)$ be a magma such that $a*(b*c)=(c*a)*b \space;\forall a,b,c\in S $ , then is it consistent that $(S,*)$ has neither ...
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More Information about Magmas

I first learnt of magmas on Wikipedia and have been trying to read more on them just out of my own interest. Whenever I try to search them on Google, though, the search results are overwhelmed by the ...
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An interesting question from “Group Theory: A First Journey”

I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page http://www.cmi.ac.in/~vipul/mathjourneys/ . In this manuscript the author proposes ...
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Does left-cancellativity of an element imply that in a superstructure, it has a left inverse?

Let $X$ denote a magma and suppose $x \in X$. Then clearly, if there exists a magma $Y$ and an injective homomorphism $f : X \rightarrow Y$ such that $f(x)$ has a left-inverse, then it follows that ...
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Is there a standard name for a set equipped only with an idempotent binary operation?

Is there a name for an idempotent magma, or do they not arise often enough to warrant a special name? (By idempotent binary operation, I mean an operation $+$ such that $x + x = x$ for any $x$.)
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What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the ...
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How to describe free magmas in more structuralist terms?

Given a generating set $G$ (assume for simplicitly it consists entirely of urelements), the free magma on $G$ can be described concretely as follows. Its underlying set is the least $U \supseteq G$ ...
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454 views

Proving that $f(A+B)=f(A)+f(B).$

Let $X$ and $Y$ denote magmas, and suppose $f : X \rightarrow Y$ is homomorphism. Then I think that for all $A,B \subseteq X$, we have $f(A+B)=f(A)+f(B).$ However, I'm not happy with my: Proof. The ...
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1answer
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Are there interesting examples of medial non-commutative semigroups?

There exist semigroups $S$ (written additively) such that $S$ is medial, meaning $(a+b)+(a'+b') = (a+a')+(b+b')$. $S$ is not commutative. Example. The left (and right) zero semigroups are all ...
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A question on certain “magma” structures

Let $S$ be a non-empty set and $*$ be a binary operation on $S$ such that the following axioms hold: for every $x,y,z∈S$ , $x*(y*z)=(x*z)*y$; $(S,*)$ has a right identity i.e. there exists $e∈S$ ...
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1answer
48 views

Is there a name for magmas with $[x+y]+[x'+y'] \equiv [x+x']+[y+y']$?

Is there a name for magmas (written additively) satsisfying the following identity? The square brackets have no particular signifance, but will hopefully promote readability in what follows. ...
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30 views

Proper term for a monotonic magma

Let $M$ be the multiplication table for a finite magma. Entries are labeled $0,\ldots,n-1$. M has the property that $M(i,j)\ge i, M(i,j) \ge j$. What is the proper term for this kind of magma and ...
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How are the powers being changed

I have a semigroup $S$ including a generator, say $d$, such that $$d^4=d$$ I am trying to guess the general rule of $d$'s powers such that when I want to calculate $d^n, n\in\mathbb N$; I can simplify ...
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Inverse elements in the absence of identities/associativity.

Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows. $(xy)z=x(yz).$ $xe=ex=x.$ ...
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Which properties are inherited by the Cartesian product of two sets equipped with a binary operation?

Let $G$ and $H$ denote sets equipped with a binary operation (aka magmas). We can form the Cartesian product magma $G \times H$ in the obvious way. I'm interested in which properties of $G$ and $H$ ...
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Left continuous magmas with no fixed points

Let $X$ be a compact Hausdorff topological space, and $*: X^2\rightarrow X$ an associative map (so that $(X, *)$ is a semigroup) which is left continuous (for all $s\in X$, the map $t\mapsto ts$ is ...
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Associativity for Magma

Say I have the operation table for a magma. I want to know whether or not the operation is associative. However, associativity is defined for an operation on 3 elements, and the operation table deals ...
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Prove this magma commutativity

Given magma $(X, *)$ and that $(x*y)*y = x$ and $y*(y*x) = x$ $\forall x, y \in X$ prove that $x*y = y*x$. Should be rather simple but I've been trying to prove that for several hours now with no ...
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Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...
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Properties of finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$?

I am considering finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$. Any finite commutative group is an example of such thing. But in the context (this question ...