Algebra of Normal Forms Is a Heyting Algebra1

We prove that the lattice of normal forms over an arbitrary set, introduced in [12], is an implicative lattice. The relative pseudo-complement α ⇒ β is defined as α 1 ∪α 2 =α −α 1 α 2 β, where −α is the pseudo-complement of α and α β is a rather strong implication introduced in this paper. [1] provide the notation and terminology for this paper. The following proposition is true (1) Let A, B, C be non empty sets and f be a function from A into B. Suppose that for every element x of A holds f (x) ∈ C. Then f is a function from A into C. In the sequel A is a non empty set and a is an element of A. Let us consider A and let B, C be elements of Fin A. Let us observe that B ⊆ C if and only if: (Def. 1) For every a such that a ∈ B holds a ∈ C. Let A be a non empty set and let B be a non empty subset of A. Then B → is a function from B into A. In the sequel A denotes a set. Let us consider A. Let us assume that A is non empty. The functor [A] yields a non empty set and is defined as follows: (Def. 2) [A] = A. We adopt the following rules: B, C are elements of Fin DP(A), a, b, c, s, t 1 , t 2 are elements of DP(A), and u, v, w are elements of the lattice of normal forms over A. Next we state the proposition (3) 1 If B = / 0, then µB = / 0. Let us consider A. Observe that there exists an element of the normal forms over A which is non empty. Let us consider A, a. Then {a} is an element of the normal forms over A. Let us consider A and let u be an element of the lattice of normal forms over A. The functor @ u yielding an element of the normal forms over A is defined by: 1 The proposition (2) has been removed.