Algebra of Normal Forms

We mean by a normal form a finite set of ordered pairs of subsets of a fixed set that fulfils two conditions: elements of it consist of disjoint sets and elements of it are incomparable w.r.t. inclusion. The underlying set corresponds to a set of propositional variables but is arbitrary. The correspodents to a normal form of a formula, e.g. a disjunctive normal form, is as follows. The normal form is the set of disjuncts and a disjunct is an ordered pair consisting of the sets of propostional variables that occur in the non-negated and negated disjunct. The requirement that the element of a normal form consists of disjoint sets means that contradictory disjuncts have been removed, and the second condition means that the absorption law has been used to shorten the normal form. We construct a lattice 〈 , ⊔,⊓〉 , where a⊔b = μ(a∪b) and a ⊓ b = μc, c being the set of all pairs 〈X1 ∪ Y1,X2 ∪ Y2〉, 〈X1,X2〉 ∈ a and 〈Y1, Y2〉 ∈ b, which consist of disjoint sets. μa denotes here the set of all minimal, w.r.t. inclusion, elements of a. We prove that the lattice of normal forms over a set defined in this way is distributive and that ∅ is the minimal element of it.