# quasi ideal in semigroup

I am new in semigroup theory and I have an problem related to quasi ideals. I am trying to solve the following problem

Problem: For a semigroup $$S$$, the following are equivalent

1. $$S$$ is regular

2. $$A=ASA$$ for every quasi ideal $$A$$

My approach for $$1\implies 2$$

Consider $$a\in A$$ since $$S$$ is regular so there exist $$x\in S$$ such that $$a=axa$$ so $$A\subseteq ASA$$

But I am not able to prove $$ASA\subseteq A$$

Any help in this regard

EDIT. According to Clifford and Preston (The algebraic theory of semigroups, Volume I, p. 85), a subset $$A$$ of $$S$$ is a quasi-ideal if $$AS \cap SA \subseteq A$$.

• Well, for starters, you are supposed to prove that $ASA\subseteq A$, not that $SAS\subseteq A$. Jan 30, 2020 at 17:40
• Note that $asa=(as)a = a(sa)\in SA\cap AS$. Jan 30, 2020 at 17:42
• oh yes,my mistake
– kam
Jan 30, 2020 at 17:42
• @ArturoMagidin Thanks! i get it
– kam
Jan 30, 2020 at 17:46
• Could you please give the definition of a quasi ideal of a semigroup? Feb 1, 2020 at 6:50

Let $$A$$ be a quasi-ideal of $$S$$. Since $$SA \subseteq S$$ and $$AS \subseteq S$$, one gets $$ASA \subseteq AS \cap SA \subseteq A$$.