Green relations in semigroups : how to interpret $J(x) \setminus J_x$

Question

I read in semigroup theory that given a semigroup $S^1$ (which has an identity), the $\mathcal{J}$ Green relation has an associated function $J(x)$:

$$ J(x) = S^1xS^1 $$

which is the principal ideal generated by $x$ given the Green relation $\mathcal{J}$. On the other hand, $J_x$ refers to the equivalence class of $x$, i.e. $x\mathcal{J}y \implies S^1xS^1 = S^1yS^1$, so that $y \in J_x$.

But how do I interpret set $I(x)$, which is:

$$ I(x) = J(x) \setminus J_x $$

From what I read, $J(x)/I(x)$ is the principal factor of semigroup $S^1$.

score 1 · Accepted Answer · answered Jun 03 '19 at 05:04

1

$I(X)$ is the set of elements that are strictly $\mathcal{J}$-below $x$: $$ I(x) = \{y \in S \mid y <_\mathcal{J} x \} $$

answered Jun 03 '19 at 05:04

J.-E. Pin

42,871

Thanks, just a short follow-up question... does that imply that $$J_x = {y \in S | y \geq_{\mathcal{J}} x }$$ ? – Link L Jun 03 '19 at 05:27
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No, $$J_x = {y \in S \mid y \leqslant_\mathcal{J} x }$$ – J.-E. Pin Jun 03 '19 at 06:20

Green relations in semigroups : how to interpret $J(x) \setminus J_x$

1 Answers1