Boolean Metric Spaces and Boolean Algebraic Varieties

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps An −→ Am in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e. sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces. Notations and conventions. Throughout this work, (B,+, ·) will be a Boolean ring where the operation a∨ b = a+ b+ ab is the analogue for set union, the order a ≤ b ⇔ ab = a is the analogue for set inclusion and for each a ∈ B, ā = a + 1 is the analogue for the set complement of a. All rings will be commutative with identity. Regular ring will mean here commutative Von Neumann regular ring, i.e. a (commutative) ring for which any principal ideal is generated by an idempotent, also known as absolutely flat rings, see [6], [12]. Unless otherwise stated, A will be a regular ring, B(A) will denote the set of the idempotent elements of A and e : A −→ B(A) will be the map that sends each a ∈ A to the only idempotent e(a) ∈ B(A) such that aA = e(a)A. The set B(A) has a structure of Boolean ring with product inherited from A and with the sum a+̃b = (a − b). For a1, . . . , an ∈ B(A) with aiaj = 0 for i 6= j, it holds a = a1 + · · · + an = a1+̃ · · · +̃an = a1 ∨ · · · ∨ an. In this case we will denote a by a1 ⊕ · · · ⊕ an. Given a prime p ∈ Z, a p-ring is a ring A for which px = 0 and x = x for all x ∈ A. In particular, a Boolean ring is a 2-ring. Any p-ring is a regular ring with e(x) = x. An algebraic variety over a ring A is a set U ⊂ A which is the set of solutions to a finite number of polynomial equations. If U ⊂ A and V ⊂ A are algebraic varieties, a map f : U −→ V is called a polynomial map if there are polynomials f1, . . . , fm ∈ A[X1, . . .Xn] such that f(x) = (f1(x), . . . , fm(x)). When A = B is a Boolean ring the usual terms are Boolean domain and Boolean transformation, see [13] and [14]. Introduction. Boolean metric spaces (Definition 1.1) appeared in several works in the 1950’s and 1960’s [2], [3], [4], [5], [7] and [8], where some authors investigated the 2000 Mathematics Subject Classification. MSC Primary: 13AXX. MSC Secondary: 06E30, 16E50, 51F99. Author supported by FPU grant of SEEU-MECD, Spain. 1