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 K, the poset RE_K (T) of weak-relative closure spaces is a sublattice of CSP(X) (the family of all the spaces with support X). On the other hand, it will be shown that the poset 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.