On lattices and algebras of simple functions
Let be the general Boolean algebra and a linear operator on . If for any in (, respectively), is regular (invertible, respectively) if and only if is regular (invertible, respectively), then is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over . Meanwhile, noting that a general Boolean algebra is isomorphic...
Among completely regular locales, we characterize those that have the feature described in the title. They are, of course, localic analogues of what are called -isocompact spaces. They have been considered in T. Dube, I. Naidoo, C. N. Ncube (2014), so here we give new characterizations that do not appear in this reference.
We construct algebras of sets which are not MB-representable. The existence of such algebras was previously known under additional set-theoretic assumptions. On the other hand, we prove that every Boolean algebra is isomorphic to an MB-representable algebra of sets.