We show that prohibiting a combinatorial tree in the Priestley duals determines an axiomatizable class of distributive lattices. On the other hand, prohibiting -crowns with does not. Given what is known about the diamond, this is another strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.
The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush in 1990. We complete this algebra with involutions in its sections and show that it can be converted into a certain implication algebra. Asking several additional conditions, we show whether this directoid is sectionally complemented or whether the section is an NMV-algebra.
Posets with property DINT which are compact pospaces with respect to the interval topologies are characterized.