Showing that (T,*) is a semigroup where T is the trace and * is defined inside

Question

Let $D$ be a $D−class$ of a semigroup $S$. The Trace of $D$ is $T = D ∪ {0}$ where $0$ is a symbol not in $D$. Define a binary operation $∗$ on $D$ by:

$a ∗ b = ab$ for $a, b ∈ D$ and $ ab ∈ R_a ∩ L_b$

$a*b = 0$ otherwise

Show that $(T, ∗)$ is a semigroup.

I am having issues showing associativity for this! Any help would be much appreciated.

Answer 1

Hint. Use the fact (due to Miller and Clifford 1956) that $ab \in R_a \cap L_b$ if and only if $R_b \cap L_a$ contains an idempotent.