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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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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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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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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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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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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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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Generalization of prime and irreducible elements to arbitary magmas
Has anyone ever given a general definition of prime and irreducible elements in arbitrary magmas, that is to say, sets with a single binary operation with no restrictions? I know Walter Noll has given ...
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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 $iter(...
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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 $x_1\cdot(x_2\cdot ...
user123733
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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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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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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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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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Are there useful visual representations of magmas?
In group theory we have Cayley graphs. Are there analogous or anyway useful visual representations of magma structures?
I am unsure about how to construct a graph representing, for instance, a free ...
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Does the following property of the composition of a magma have a name?
If $M$ is a magma and
$$+:M\times M\to M$$
is its law of composition, does the property
$$(x+y)+z=x+(y+z)\qquad\forall\ x,y,z\in M :\quad y\neq x,z$$
have a name? It resembles the associativity of ...
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