Maximal d-Elements of an Algebraic Frame

Abstract

The space of maximal d-ideals of C(X) is well-known and is widely studied. It is known that the space of maximal d-ideals is homeomorphic to the Z♯(X)-ultrafilters, and this space is the minimal quasi F-cover of a compact Tychonoff space X. In the current article we generalize this concept for M-frames, algebraic frames with the finite intersection property. In particular, we explore various properties of the maximal d-elements of a frame L, and their relation with the ultrafilters of \(\mathfrak {K}L^{\perp }\), the polars of the compact elements of L. On a separate note, we revisit the Lemma on Ultrafilters and establish the correspondence between the minimal prime elements spaces of L with the spaces of ultrafilters of \(\mathfrak {K}L\). Finally, we show that for complemented frames L, Min(L) = Max(dL), a result parallel to the one known for Riesz spaces, W-objects, and topological spaces.