Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).
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 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 ⊤).
This means that for any a in A:
join(zero, a) = a = meet(one, a)
Or written using traditional notation:
(0 ∨ a) = a = (1 ∧ a)
Heyting algebras are bounded lattices that are also equipped with
an additional binary operation imp
(for impliciation, also
written as →).
Heyting algebras are bounded lattices that are also equipped with
an additional binary operation imp
(for impliciation, also
written as →).
Implication obeys the following laws:
In heyting algebras, and
is equivalent to meet
and or
is
equivalent to join
; both methods are available.
Heyting algebra also define complement
operation (sometimes
written as ¬a). The complement of a
is equivalent to (a → 0)
,
and the following laws hold:
However, in Heyting algebras this operation is only a pseudo-complement, since Heyting algebras do not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1.
Heyting algebras model intuitionistic logic. For a model of
classical logic, see the boolean algebra type class implemented as
Bool
.
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 lattice is a set A
together with two operations (meet and
join). Both operations individually constitute semilattices (join-
and meet-semilattices respectively): each operation is commutative,
associative, and idempotent.
The join and meet operations are also linked by absorption laws:
meet(a, join(a, b)) = join(a, meet(a, b)) = a
Additionally, join and meet distribute:
Join can be thought of as finding a least upper bound (supremum), and meet can be thought of as finding a greatest lower bound (infimum).
In many texts the following symbols are used:
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.
Boolean algebras are Heyting algebras with the additional constraint that the law of the excluded middle is true (equivalently, double-negation is true).
This means that in addition to the laws Heyting algebras obey, boolean algebras also obey the following:
Boolean algebras generalize classical logic: one is equivalent to "true" and zero is equivalent to "false". Boolean algebras provide additional logical operators such as
xor
,nand
,nor
, andnxor
which are commonly used.Every boolean algebras has a dual algebra, which involves reversing true/false as well as and/or.