6
votes
0answers
50 views

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 ...
4
votes
1answer
60 views

Are there interesting examples of medial non-commutative semigroups?

There exist semigroups $S$ (written additively) such that $S$ is medial, meaning $(a+b)+(a'+b') = (a+a')+(b+b')$. $S$ is not commutative. Example. The left (and right) zero semigroups are all ...
1
vote
0answers
30 views

Proper term for a monotonic magma

Let $M$ be the multiplication table for a finite magma. Entries are labeled $0,\ldots,n-1$. M has the property that $M(i,j)\ge i, M(i,j) \ge j$. What is the proper term for this kind of magma and ...
3
votes
3answers
117 views

How are the powers being changed

I have a semigroup $S$ including a generator, say $d$, such that $$d^4=d$$ I am trying to guess the general rule of $d$'s powers such that when I want to calculate $d^n, n\in\mathbb N$; I can simplify ...
1
vote
4answers
212 views

Inverse elements in the absence of identities/associativity.

Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows. $(xy)z=x(yz).$ $xe=ex=x.$ ...
13
votes
5answers
1k views

Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...