All Questions
Tagged with magma abstract-algebra
125
questions
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64
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Is subtraction on the reals isomorphic to division on the positive reals?
I know that the magma $(\mathbb{R};+)$ of addition on the real numbers is isomorphic to the magma $(\mathbb{R}^+;\times)$ of multiplication on the strictly positive real numbers. I wonder, is it the ...
- 21.5k
2
votes
0
answers
28
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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\...
-4
votes
1
answer
79
views
Practical example of differences between associativity and alternativity (and the in-between Bol loop)? [closed]
Associativity is:
$$(a * b) * c = a * (b * c)$$
Alternativity is:
$$a * (a * b) = (a * a) * b$$
$$(a * b) * b = a * (b * b)$$
Bol loop is:
$${\displaystyle a(b(ac))=(a(ba))c}$$
$${\displaystyle ((ca)b)...
- 3,773
3
votes
1
answer
71
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Are there 45 unital magmas with three elements (up to isomorphism)?
How many unital magmas (magma with an identity element) with three elements are there (up to isomorphism)?
My approach:
List out all of the possible 2x2 multiplication tables for the two non-identity ...
- 135
20
votes
5
answers
3k
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Why did I never learn about magmas?
While I’ve never taken an actual abstract algebra course, there are some things I know about the typical curriculum structure:
First, define an algebraic structure.
Explain groups.
Everything else.
...
user1286095
0
votes
0
answers
82
views
Nomenclature for a unital magma together with a monoid
Is there some established name/nomenclature for structures $\mathfrak{A} = (A,\, {\oplus},\, {\odot})$, where
$(A,\, {\oplus})$ forms a (commutative) unital magma (in particular not associative!),
$(...
- 281
0
votes
1
answer
20
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Maximal Extension Chain of Halfgroupoids
A book I am reading gives the following definitions:
A collection $\{L_i:i=0,1,2,...\}$ of halfgroupoids $L_i$ is called an extension chain if $L_{i+1}$ is an extension of $L_i$ for each $i$. If $G$ ...
- 31
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34
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Closest Equivalent to Cayley Graphs for Partial Groupoids?
[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.]
This question may be nonsensical, given that the duality ...
- 31
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votes
1
answer
43
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Does 2nd power idempotency imply all nth powers idempotency?
Suppose $(M,*)$ is a magma, that is, just a set with a binary operation with no conditions imposed, and let $s$ be an element of $M$. Also, let $n$ be an integer greater than or equal to $2$. An $n$-...
- 21.5k
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1
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43
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Does there exist a magma where every element has a left cube root but not every element has a right cube root?
Let $(M,*)$ be a magma. $x$ is said to be a left cube root of $y$ if $(x*x)*x=y$. $x$ is said to be a right cube root of $y$ if $x*(x*x)=y$. Does there exist a magma where every element has a left ...
- 21.5k
3
votes
0
answers
213
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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 ...
- 20.8k
0
votes
1
answer
94
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Eckmann–Hilton Argument and magma homomorphisms
The Eckmann-Hilton result is as follows:
Let $X$ be a set equipped with two binary operations $\circ$ and $\otimes$, and suppose
$\circ$ and $\otimes$ are both unital, meaning there are identity
...
- 990
3
votes
1
answer
124
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Which axiom can almost determine the magma with one element?
The axiom $((a * b) * c) * (a * ((a * c) * a)) = c$ uniquely determines Boolean algebra, an example of a single axiom giving a magma an "interesting" structure. What is the fewest number of ...
- 4,057
1
vote
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78
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non-commutative algebraic structure with 16 elements, need help categorizing it and finding a representation
We have an abstract algebraic structure with the following multiplication table, has anyone seen this structure before and can anyone give it a proper name and a simple (possibly matrix) ...
- 21
3
votes
1
answer
100
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Is there a concept representing "connectedness" in abstract algebra?
Consider an object, call it a web, that consists of a set $S$ equipped with a binary operation obeying these axioms:
$$
\forall\ a,b \in S\ \exists\ c \in S :a\ \bullet\ b=c
$$
$$
\forall\ a,b \in S\ \...
- 31