4
votes
1answer
31 views

Semigroup which satisfied $(xy)^{\pi} = (xy)^{\pi}x$ and principal right ideals

Let $(S,\cdot)$ be a finite semigroup, then every $s \in S$ has an unique idempotent power, i.e. there exists a smallest $i \in \mathbb N$ such that $s^i$ is idempotent. It is the unique idempotent in ...
1
vote
1answer
163 views

Find identity element, invertible and inverses in $T=\mathbb Z \times \mathbb Q$

Let the following operation be defined on $T=\mathbb Z \times \mathbb Q$: $$\begin{aligned}(a,b)\centerdot (c,d) = (-ac, b+d+2) \end{aligned}$$ in the commutative semigroup $(T, \centerdot)$, find ...
1
vote
0answers
72 views

In a semigroup $S=\{a_1, \ldots, a_n\}$, any product $a_1 * \ldots * a_i, 1 \le i \le n$ is unique [duplicate]

Possible Duplicate: How does one actually show from associativity that one can drop parentheses? Claim: Let $(S, *)$ be a semigroup and $a_1, \ldots, a_n \in S$. Then $a_1 * \ldots * a_n$ ...