In this note we give some new characterizations of distributivity of a nearlattice and we study annihilator-preserving congruence relations.

The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra $H$ considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice $DedH$ of all deductive systems on $H$ and every maximal deductive system is prime. Complements and relative complements of $DedH$ are characterized as the so called annihilators in $H$.

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra $𝒜$. We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice $𝒟\left(A\right)$ of all deductive systems on $𝒜$. Moreover, relative annihilators of $C\in 𝒟\left(A\right)$ with respect to $B\in 𝒟\left(A\right)$ are introduced and serve as relative pseudocomplements of $C$ w.r.t. $B$ in $𝒟\left(A\right)$.

A new ⋄-like principle ${\diamond }_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg {\diamond }_{}$ is consistent with CH and that in many models of = ω₁ the principle ${\diamond }_{}$ holds. As ${\diamond }_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ${\diamond }_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

It is consistent that $${\aleph }_1<\mathrm{add}\left(𝒩\right)<\mathrm{add}\left(ℳ\right)=𝔟<\mathrm{cov}\left(𝒩\right)<\mathrm{non}\left(ℳ\right)<\mathrm{cov}\left(ℳ\right)=2^{{\aleph }_0}.$$ Assuming four strongly compact cardinals, it is consistent that $$\begin{array}{ccc}\hfill {\aleph }_1& <\mathrm{add}\left(𝒩\right)<\mathrm{add}\left(ℳ\right)=𝔟<\mathrm{cov}\left(𝒩\right)<\mathrm{non}\left(ℳ\right)\hfill & \hfill <\mathrm{cov}\left(ℳ\right)<\mathrm{non}\left(𝒩\right)<\mathrm{cof}\left(ℳ\right)=𝔡<\mathrm{cof}\left(𝒩\right)<2^{{\aleph }_0}.\end{array}$$

Shelah’s pcf theory describes a certain structure which must exist if ${\aleph }_{\omega }$ is strong limit and $2^{{\aleph }_{\omega }}>{\aleph }_{{\omega }_1}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially...