All Questions
Tagged with magma semigroups
36
questions
0
votes
1
answer
55
views
Term for a Set Equipped With a Binary Operation Which Contains Inverses
Let $A$ be a set and let $\circ:A\times A\rightarrow B,$ $A\subseteq B$ be a binary operation ($A$ is not necessarily closed under $\circ$). If there exists some unique $e\in A$ such that $e\circ a=a\...
- 491
3
votes
0
answers
213
views
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
2
votes
1
answer
176
views
How to show that a compact semigroup for which the cancellation law holds is a compact group
Here is my problem:
Set $G$ a compact semigroup (that is a Hausdorff compact space endowed with an associative continuous binary operation). Assume that the cancellation law holds i.e. for any $g,h,k \...
- 166
1
vote
1
answer
99
views
Term for a semigroup with left identities and left inverses?
Is there a term for a semigroup $(M, *)$ that has at least one
left identity and left inverses in the "weak" sense that, for all
$a \in M$, there exists a $b \in M$ such that $b*a$ is a left
...
3
votes
2
answers
169
views
How many non-isomorphic algebraic structures (i.e. magmas, monoids, groups etc.) are there with countably infinite order? [closed]
For structures of finite order it seems obvious to me that there are countably infinite in total, by a simple diagonalization argument (starting at all of order 1, then 2 etc.). It is however not ...
- 462
1
vote
1
answer
118
views
if $\cdot$ and $\odot$ are associative operations on $\mathbb{Z}$ when is the sum $(\cdot + \odot)$ associative?
Where $a(\cdot + \odot)b$ is defined as $(a\cdot b) + (a\odot b)$.
I know if $\cdot$ and $\odot$ distribute through addition (i.e. $a\cdot(b+c)=a\cdot b+ a\cdot c$) then the sum $(\cdot + \odot)$ is ...
- 822
5
votes
2
answers
498
views
Good book for self-study of Magmas/Semigroups/etc.?
I'm currently an undergrad in my second semester of Abstract Algebra. We've covered groups, rings, fields, all that fun stuff. I'm working with Shahriari's "Algebra in Action" as well as ...
5
votes
0
answers
204
views
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_-...
- 274
2
votes
1
answer
238
views
Fill in a partly filled in table such that it makes the magma $(M,*)$ associative, commutative, has an identity element and has no zero-elements.
Below is a partly filled in table for a binary operation ($*$) on the set $M=\{a,b,c,d\}$. I am trying to fill in the rest such that the magma $(M,*)$ becomes associative, commutative, has an identity ...
- 417
3
votes
7
answers
122
views
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{...
- 189
5
votes
3
answers
753
views
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 ...
- 65
0
votes
2
answers
1k
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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 ...
- 2,237
4
votes
1
answer
393
views
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.
...
- 1,120
1
vote
1
answer
102
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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 ...
- 597
6
votes
8
answers
7k
views
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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