0
$\begingroup$

Given a semigroup $S$, a subset $I$ is called an ideal iff for every $s \in S$ we have $sI, Is \subseteq I$. Further we set $$ s \le_{\mathcal J} t :\Leftrightarrow SsS \cup \{s\} \subseteq StS \cup \{t\} $$ i.e. the principal ideals generated are contained in each other. This gives a preorder, and by passing to the partial order by the usual identification $s \equiv_{\mathcal J} t$ iff $s \le_{\mathcal J} t$ and $t \le_{\mathcal J} s$ we get an equivalence relation, whose equivalence classes are called $\mathcal J$-classes, a particular Green relation.

For a given ideal $I$ we have for $s \in I$ that $[s]_{\equiv_{\mathcal J}} \subseteq SsS \cup \{s\} \subseteq I$, hence every ideal is a disjoint union of equivalence classes.

Is there any useful characterisation of when an arbitrary union of equivalence classes forms an ideal?

$\endgroup$
  • $\begingroup$ The ideals are lower sets in the J-quasiorder. So your collection of J-classes must form a lower set in S/J $\endgroup$ – Benjamin Steinberg Jun 22 '18 at 15:27

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.