Definition 1.4.1 (Lattice). A lattice is a set $L$ equipped with binary commutative and associative operations $\land$ and $\lor$ that satisfy the absorption laws:

a \lor (a \land b) = a; a \land (a \lor b) = a,

for all $a, b \in L$ .

A lattice is:

Distributive if $a \land (b \lor c) = (a \land b) \lor (a \land c)$ for all $a, b, c \in L$ .
Bounded if there are elements $⊥, ⊤ \in L$ such that $a \lor ⊥ = a$ and $a \land ⊤ = a$ .
Complemented if it is bounded and for every $a \in L$ there is $a^{*} \in L$ such that $a \land a^{*} = ⊥$ and $a \lor a^{*} = ⊤$ .

A Boolean algebra is a complemented distributive lattice.