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?

  • $\begingroup$ 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

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