Automaticity of one-relator semigroups with length less than or equal to three

The main results of this paper is to give a complete characterization of the automaticity of one-relator semigroups with length less than or equal to three. Let $S=sgp\langle A|u=v\rangle$ be a semigroup generated by a set $A=\{a_1,a_2,\dots,a_n\},\ n\in \mathbb{N}$ with defining relation $u=v$, where $u,v\in A^*$ and $A^*$ is the free monoid generated by $A$. Such a semigroup is called a one-relator semigroup. Suppose that $|v|\leq|u|\leq3$, where $|u|$ is the length of the word $u$. Suppose that $a,b\in A,\ a\neq b$. Then we have the following: (1) $S$ is prefix-automatic if $u=v\not\in \{aba=ba,\ aab=ba,\ abb=bb\}$. Moreover, if $u=v\in \{aba=ba,\ aab=ba,\ abb=bb\}$ then $S$ is not automatic. (2) $S$ is biautomatic if one of the following holds: (i) $|u|=3,\ |v|=0$, (ii) $|u|=|v|=3$, (iii) $|u|=2$ and $u=v\not\in \{ab=a,\ ab=b\}$. Moreover, if $u=v\in \{ab=a,\ ab=b\}$ then $S$ is not biautomatic.