Some properties of primal topologies seen as semirings (Q6539334)

Let \(X\) be a set and \(f \colon X \to X\) a function. According to [\textit{O. Echi}, Topology Appl. 159, No. 9, 2357--2366 (2012; Zbl 1245.54033)], the topology \(\mathcal{T}_f\) on \(X\), where the open sets are those \(O \subseteq X\) such that \(f^{-1}(O) \subseteq O\), is called a primal topology.\N\NThis paper is devoted to studying various properties of primal topologies. In certain cases, these properties can be interpreted as properties of semirings.\N\NLet me highlight some of the results that the authors have proven:\N\N\begin{itemize}\N\item[--] A primal space \((X, \mathcal{T}_f)\) is connected if and only if every continuous function from \(X\) to a \(T_1\) space \(Y\) with cardinality greater than 1 is constant.\N\item[--] If \((X, \mathcal{T}_f)\) is connected, then \(\mathcal{T}_f\) is a local semiring.\N\item[--] The authors also provided some topological conditions for a square matrix \(A\) to be invertible, considering the primal topology \(\mathcal{T}_A\) induced by the linear transformation associated with the matrix.\N\end{itemize}\N\NIt is worth noting that the paper is clear and concise, and the proofs are presented in an elegant manner.