On yosida frames and related frames

Abstract

Topological structures called Yosida frames and related algebraic frames are studied in the realm of Pointfree Topology. It is shown that in algebraic frames regular elements are those for which compact elements are rather below the regular elements, and algebraic frames are regular if and only if every compact element is rather below itself if and only if the frame has the Finite Intersection Property (FIP) and each prime element is minimal. We also show that Yosida frames are those algebraic frames with the Finite Intersection Property and are finitely subfit; that these frames are also those semi-simple algebraic frames with FIP and a disjointification where dim (L)≤ 1; and we prove that in an algebraic frame with FIP, it holds that dom (L) = dim (L). In relation to normality in Yosida frames, we show that in a coherent normal Yosida frame L, the frame is subfit if and only if it is regular if and only if it is zero- dimensional if and only if every compact element is complemented.