The spectrum of $\sigma$-frames in the adjunction between $\sigma$-frames and $\sigma$-spaces

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Go.C., Islamic Azad University, Gorgan, Iran

2 Department of Mathematics, Go.C., Islamic Azad University, Gorgan, Iran.

10.48308/cgasa.2026.241071.1565

Abstract

In this paper, we define an adjunction between two categories: $\sigma$-frames and $\sigma$-spaces, denoted by the pair $(\Sigma^\sigma, \Lambda)$. The functor $\Sigma^\sigma$ is constructed using the concept of $\sigma$-points. We prove that $\sigma$-points are equivalent to $\sigma$-completely prime filters, but unlike in pointfree topology, they do not correspond to prime elements. While every prime element determines a corresponding $\sigma$-point, the converse fails. For $\sigma$-frames, we define the $\sigma$-spatiality condition, which is equivalent to having enough $\sigma$-points. Dually, for $\sigma$-spaces, the $\sigma$-soberness condition is shown to be equivalent to the conjunction of the $\sigma_0$ separation axiom and $\sigma$-pointedness properties. These conditions naturally emerge from careful analysis of the adjunction morphisms.

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 12 May 2026

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