All Questions
6
questions
2
votes
2
answers
199
views
What is the name for a magma which is neither a quasigroup nor a semigroup yet has both an identity and inverses?
Is there a name which is more specific than `unital magma' for a magma whose only requirements are that it should have both an identity and (L/R symmetric) inverses for all elements?
The following ...
- 407
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
-1
votes
1
answer
209
views
Inverse element in a magma [closed]
Given $(S,*)$ a magma and an identity element $e$. The inverse of $x\in S$ is $y$ such that $x*y=e=y*x$.
Is it correct to say that if $x$ is the inverse of $y$ then $y$ is the inverse of $x$?
- 11
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
0
votes
1
answer
38
views
Is "(a * a') is cancellative" + "M has an identity" the same as "a has an inverse"
Given a magma $(M, \ast)$, $(a \ast a')$ is cancellative, iff
$$\forall b,c \in M. b \ast (a \ast a') = c \ast (a \ast a')\Leftrightarrow b = c$$
The magma has an identity, iff:
$$\exists e.\forall ...
- 1,257
2
votes
1
answer
88
views
Finding a binary operation on $\{1, \dots, n\}$ so that each $k$ has exactly $k - 1$ left inverses
What is an example of a binary operation on the set $\{1, \dots, n\}$ so that each element $k \in \{1, \dots, n\}$ has respectively $k-1$ left inverses?
I have been trying various combinations with $...
- 323