All Questions
Tagged with magma quasigroups
8
questions
-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
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
5
votes
1
answer
160
views
Do the Moufang identities *themselves* imply diassociativity / Moufang's theorem / Artin's theorem?
A Moufang loop is a loop satisfying the Moufang identities. Famously, these are diassociative -- the subloop generated by any two elements is associative (is a group) -- and more generally, they ...
- 4,742
1
vote
1
answer
233
views
Inverse element of a magma
It is accepted that two elements are inverse to each other if their product is equal to the identity element:
Inverse element in a magma
https://en.wikipedia.org/wiki/Inverse_element
The definition ...
- 1,120
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
7
votes
2
answers
551
views
Defining loops: why is divisibility and identitiy implying invertibility?
Wikipedia contains the following figure (to be found, e.g. here) in order to visualize the relations between several algebraic structures. I highlighted a part that I find especially interesting.
It ...
- 30.2k
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
0
answers
54
views
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 ...
- 11