$M_K(T)$-Spaces: Some Order-Theoretical Properties and Applications to Distributive Logics - Brunetta, Cristian; Fernández, Víctor (Armenian Journal of Mathematics, 2024)
We present some results involving several ordered structures related to the construction of $M_K(T)$-spaces. These are special closure spaces defined on the basis of a previously given one ${K}$ and were developed by Fern\'andez and Brunetta, 2023. Specifically, we show that for every $T \in {K}$, the poset $\mathbb{RE}_{K}(T)$ of weak-relative closure spaces is a sublattice of $\mathbb{CSP}(X)$ (the family of all the spaces with support $X$). On the other hand, it will be shown that the poset $\mathbb{M}(K)$ of all the $M_{K}(T)$-spaces does not verify this property, but it is a complete lattice itself. Also, we show in which way some order-theoretic properties are related to the recovery of closure spaces. Finally, we show some applications of $M_{K}(T)$-spaces to the class of distributive logics.