A question about a confluent abstract rewriting system? Let $(A,\rightarrow)$ be a confluent abstract rewriting system. Assume that $a\stackrel{*}\rightarrow b$, $a\stackrel{*}\rightarrow c$ is a fork. Then $b,c$ is joinable, that is, there is a $d\in A$ such that $b\stackrel{*}\rightarrow d$, $c\stackrel{*}\rightarrow d$. My question is whether $d\neq b$ and $d\neq c$?

No, $d$ need not be different from both $b$ and $c$. For instance, if $A = \\{a,b,c\\}$, the rewrite system $(A, \\{ a \rightarrow b,\, a \rightarrow c,\,b \rightarrow c\\})$ is confluent, but the only way to join $b$ and $c$ is $b \rightarrow^1 d \leftarrow^0 c$ with $d$ identical to $c$. Remember that $x \rightarrow^* y$ means that there exists some $n \geq 0$ (!) with $x \rightarrow^n y$, so the case $n = 0$ is included.