An extremally disconnected space is called an absolute retract in the class of all extremally disconnected spaces if it is a retract of any extremally disconnected compact space in which it can be embedded. The Gleason spaces over dyadic spaces have this property. The main result of this paper says that if a space X of π-weight ${\omega }_1$ is an absolute retract in the class of all extremally disconnected compact spaces and X is homogeneous with respect to π-weight (i.e. all non-empty open sets have the same...

We partially strengthen a result of Shelah from [Sh] by proving that if $\kappa ={\kappa }^{\omega }$ and $P$ is a CCC partial order with e.g. $\left|P\right|\le {\kappa }^{+\omega }$ (the ${\omega }^{\text{th}}$ successor of $\kappa$) and $\left|P\right|\le 2^{\kappa }$ then $P$ is $\kappa$-linked.

We study the minimal prime elements of multiplication lattice module $M$ over a $C$-lattice $L$. Moreover, we topologize the spectrum $\pi \left(M\right)$ of minimal prime elements of $M$ and study several properties of it. The compactness of $\pi \left(M\right)$ is characterized in several ways. Also, we investigate the interplay between the topological properties of $\pi \left(M\right)$ and algebraic properties of $M$.

The functor taking global elements of Boolean algebras in the topos $\mathrm{𝐒𝐡}𝔅$ of sheaves on a complete Boolean algebra $𝔅$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $𝔅$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.