The study of finite semigroups has its roots in Theoretical Computer Science. In particular, in the mid nineteen seventies, Eilenberg~\cite{MR0530383} established the link between ``varieties of rational languages'', which is an important object of study in Computer Science, and certain classes of finite semigroups, known as ``pseudovarieties''.

At the level of pseudovarieties, some problems arise naturally, one of them being the so-called ``word problem''. Roughly speaking, it consists in deciding whether two expressions define the same element in every semigroup of a given pseudovariety.

In this talk, we start by introducing some basic background on finite semigroups. Our first goal is to explain what the ``$\kappa$-word problem over $DRG$'' is about. After that, based on some illustrative examples, we intend to give intuition on how to show that the referred problem is decidable. Our solution extends work of Almeida and Zeitoun~\cite{MR2289710} on the pseudovariety consisting of all ``$\mathcal{R}$-trivial semigroups''.

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\bibitem{MR2289710}

J.~Almeida and M.~Zeitoun, \emph{An automata-theoretic approach to the word problem for {$\omega$}-terms over~{$\sf{R}$}}, Theoret. Comput. Sci. \textbf{370} (2007), no.~1-3, 131--169.

\bibitem{MR0530383}

S.~Eilenberg, \emph{Automata, languages, and machines. {V}ol. {B}}, Academic Press, New York-London, 1976.