Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).
Every Boolean ring gives rise to a Boolean algebra:
A bounded distributive lattice is a lattice that both bounded and distributive
A bounded lattice is a lattice that additionally has one element that is the bottom (zero, also written as ⊥), and one element that is the top (one, also written as ⊤).
A distributive lattice a lattice where join and meet distribute:
Generalized Boolean algebra, that is, a Boolean algebra without the top element.
Every Boolean rng gives rise to a Boolean algebra without top:
Heyting algebras are bounded lattices that are also equipped with
an additional binary operation imp
(for impliciation, also
written as →).
A join-semilattice (or upper semilattice) is a semilattice whose operation is called "join", and which can be thought of as a least upper bound.
A lattice is a set A
together with two operations (meet and
join).
A meet-semilattice (or lower semilattice) is a semilattice whose operation is called "meet", and which can be thought of as a greatest lower bound.