Some Properties of Primal Topologies Seen as Semirings
Keywords:
invertible matrices, primal topologies, semiringsAbstract
Given a set $X$ and a function $f: X \rightarrow X$, a topology $\tau_f$ is determined by taking the open sets to be those sets $A \subset X$ such that $f^{-1}(A) \subseteq A$. The topological space ( $X, \tau_f$ ) is called primal space or functional Alexandroff space. In this paper, we study some properties of primal topologies seen as semirings. We prove, for example, that if ( $X, \tau_f$ ) is a connected primal space, then $\tau_f$ is a local semiring. We also determine some topological conditions for a square matrix A to be invertible, considering the primal topology $\tau_A$ generated by the matrix.
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