Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring ( a Boolean ring ) or a special kind of distributive lattice ( a Boolean lattice ).
22.
Given a Boolean algebra " B ", we turn " B " into a Boolean ring by using the symmetric difference as addition and the meet operation \ land as multiplication.
23.
Historically, the term " Boolean ring " has been used to mean a " Boolean ring possibly without an identity ", and " Boolean algebra " has been used to mean a Boolean ring with an identity.
24.
Historically, the term " Boolean ring " has been used to mean a " Boolean ring possibly without an identity ", and " Boolean algebra " has been used to mean a Boolean ring with an identity.
25.
Historically, the term " Boolean ring " has been used to mean a " Boolean ring possibly without an identity ", and " Boolean algebra " has been used to mean a Boolean ring with an identity.
26.
He extensively studied the role of duality in Boolean theory and subsequently developed a theory of n-ality for certain rings which played for n-valued logics the role of Boolean rings vis-a-vis Boolean algebras.
27.
Furthermore, a subset of a Boolean ring is a ring ideal ( prime ring ideal, maximal ring ideal ) if and only if it is an order ideal ( prime order ideal, maximal order ideal ) of the Boolean algebra.
28.
Unification in Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise ( i . e . if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a most general unifier, and otherwise the minimal complete set of unifiers is finite ).
29.
Unification in Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise ( i . e . if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a most general unifier, and otherwise the minimal complete set of unifiers is finite ).
30.
On the other hand there is a sense in which Boolean algebras and Boolean rings can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely; in this sort of situation the homomorphisms between corresponding structures are the same.
