In Chajda's paper (2014), to an arbitrary BCI-algebra the author assigned an ordered structure with one binary operation which possesses certain antitone mappings. In the present paper, we show that a similar construction can be done also for pseudo-BCI-algebras, but the resulting structure should have two binary operations and a set of couples of antitone mappings which are in a certain sense mutually inverse. The motivation for this approach is the well-known fact that every commutative BCK-algebra...

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 $𝒟\left(A\right)$.

We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.