BOOLE-DE MORGAN ALGEBRAS AND QUASI-DE ...

BOOLE-DE MORGAN ALGEBRAS AND QUASI-DE MORGAN FUNCTIONS Yu. M. Movsisyan and V. A. Aslanyan Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, Yerevan 0025, Armenia E-mail: [email protected] E-mail: [email protected]

Abstract. In this paper we establish a Stone-type and a Birkhoff-type representation theorems for Boole-De Morgan algebras and prove that the free Boole-De Morgan algebra on n free generators is isomorphic to the Boole-De Morgan algebra of quasi-De Morgan functions of n variables. Also we introduce the concept of Zhegalkin polynomials for quasi-De Morgan functions and consider the representation problem of those functions by polynomials.

Key Words: Boole-De Morgan algebra; free algebra; Boolean function; De Morgan function; quasi-De Morgan function; disjunctive (conjunctive) normal form of quasi-De Morgan function; Zhegalkin polynomial for quasi-De Morgan function. 2010 Mathematics Subject Classification: 06B20, 06D30, 06E25, 06E30, 08B20, 08B26, 08A40.

1

Introduction and preliminaries

Let us start from the definitions of Boolean and De Morgan algebras. An algebra (Q; {+, ·, 0 , 0, 1}) with two binary, one unary and two nullary operations is called a Boolean algebra if (Q; {+, ·, 0, 1}) is a bounded distributive lattice with least element 0 and greatest element 1 and the algebra (Q; {+, ·, 0 , 0, 1}) satisfies the following identities: x + x0 = 1, x · x0 = 0. An algebra (Q; {+, ·,¯ , 0, 1}) with two binary, one unary and two nullary operations is called a De Morgan algebra if (Q; {+, ·, 0, 1}) is a bounded distributive 1

lattice with least element 0 and greatest element 1 and the algebra (Q; {+, ·,¯ }) satisfies the following identities: x + y = x · y, x = x, where x = (x) ( [1–6, 10–12, 14, 23–26]). For example, the standard fuzzy algebra F = ([0, 1]; max(x, y), min(x, y), 1 − x, 0, 1) is a De Morgan algebra. Note that in any De Morgan algebra 0 = 1, 1 = 0. For a lattice (L; {+, ·}) its partial order ≤ is defined by the following way: x ≤ y ⇔ x + y = y, x, y ∈ L. For the definition of free algebras of the given variety see [8, 9, 29, 30]. Let B = {0, 1} . Define the operations +, ·, 0 , on B by the following way: 0+0 = 0, 0+1 = 1+0 = 1+1 = 1, 0·1 = 0·0 = 1·0 = 0, 1·1 = 1, 00 = 1, 10 = 0. We get the Boolean algebra B = (B; {+, ·, 0 , 0, 1}). For a set X denote the set of all its subsets by 2X or P(X). If we consider subsets of a given set X then for a subset s ⊆ X we denote s = X \ s. A function f : B n → B is called a Boolean function of n variables, where B n is the set of all n-element sequences of B. The following result is well known. Theorem 1.1. ( [7]) For every Boolean function f : B n → B there exists a unique set S ⊆ 2{1,...,n} such that ! X Y Y f (x1 , . . . , xn ) = xi · x0i , s∈S

i∈s

i∈s

where the operations on the right hand side are the operations of Boolean algebra B. Note that those terms are called disjunctive normal forms for Boolean functions. It is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables ( [3, 8, 29]). The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables ( [3,8,29]). In paper [24] we have introduced the concept of De Morgan function and proved that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem posed by B.I. Plotkin. 2

In papers [24] we have also introduced the concept of quasi-De Morgan functions (under the name of A-functions) and used them to establish some properties of De Morgan functions. In this paper we continue the study of the properties of quasi-De Morgan functions and define disjunctive and conjunctive normal forms for those functions. Also we consider the variety of Boole-De Morgan algebras (introduced in [13, 14] under the name of Boolean bisemigroups) and prove that the free Boole-De Morgan algebra on n free generators is isomorphic to the Boole-De Morgan algebra of quasi-De Morgan functions of n variables. But first (in Section 2) we give some natural examples of Boole-De Morgan algebras and prove a Stone-type representation theorem for those algebras. Then (in Section 3) we find all subdirectly irreducible Boole-De Morgan algebras. Finally, in Section 5 we introduce the concept of Zhegalkin polynomials for the case of quasi-De Morgan functions and prove some results analogous to Zhegalkin representation theorem on Boolean functions (see [7, 32]).

2

Boole-De Morgan algebras

Definition 2.1. An algebra (Q; {+, ·,¯, 0 , 0, 1}) with two binary, two unary and two nullary operations is called a Boole-De Morgan algebra if (Q; {+, ·,¯, 0, 1}) is a De Morgan algebra and (Q; {+, ·, 0 , 0, 1}) is a Boolean algebra and the two unary operations commute, i.e., (x)0 = (x0 ). This concept is introduced in [13,14] under the name of Boolean bisemigroup. Let us consider some natural examples of Boole-De Morgan algebras. First note that every Boolean algebra can be considered as a Boole-De Morgan algebra with two equal unary operations. In particular if B = (B; {+, ·, 0 , 0, 1}) is the two-element Boolean algebra and x = x0 (i.e., the unary operations ¯ and 0 are equal) then the algebra (B; {+, ·,¯, 0 , 0, 1}) is a Boole-De Morgan algebra and we will denote it by BM 2 . Now we define a Boole-De Morgan algebra on the four-element set D = {0, a, b, 1} which will be used in the proof of the main theorem of Section 4. Defining 0 + x = x + 0 = x, 0 · x = x · 0 = 0 and 1 · x = x · 1 = x, 1 + x = x + 1 = 1 and x + x = x, x · x = x for all x ∈ D, and a + b = b + a = 1, a · b = b · a = 0, 0 = 1, 1 = 0, a = a, b = b, 10 = 0, 00 = 1, a0 = b, b0 = a we get the Boole-De Morgan algebra BM 4 = (D; {+, ·,¯, 0 , 0, 1}). For a Boolean algebra B = (Q; {+, ·, 0 , 0, 1}) consider the direct product B × B. Defining one more unary operation ¯ on the set Q × Q by (x, y) = (y 0 , x0 ) we get the Boole-De Morgan algebra B × B. Let us recall the definition of n-ary term operations. Let A = (Q; Σ) be an arbitrary algebra. The n-ary term operations (or term functions) of algebra A are 3

defined by induction: 1) all n-ary identical operations (or projections) of set Q δni (x1 , . . . , xn ) = xi , i = 1, . . . , n, are n-ary term operations of A; 2) if f1 , . . . , fm are n-ary term operations of A, then the superposition f (x1 , . . . , xn ) = f0 (f1 (x1 , . . . , xn ), . . . , fm (x1 , . . . , xn )) is again an n-ary term operation of A, for every m-ary operation f0 ∈ Σ; 3) there are no other n-ary term operations of A. An operation h on the set Q is called a term operation (function) of algebra A, if h is an n-ary term operation of A for some n. In particular, for n = 2 we get the definition of binary term operation of the given algebra A. For example, for a nontrivial lattice A the set T (A) of binary term operations of A is equal to {x + y, x · y, x, y}. The set T (A) of binary term operations of the nontrivial lattice A is a Boole-De Morgan algebra (of order 4) where the operations are defined below. For any two binary terms f (x, y) and g(x, y) the binary operations are defined as the following binary superpositions: (f + g)(x, y) = f (x, g(x, y)), (f · g)(x, y) = f (g(x, y), y). The nullary operations are the terms y and x. The unary operations are the commutation and dualization. The commutation is defined by f (x, y) = f (y, x) and for a binary term f (x, y) to get its dual term f 0 (x, y) we shall change all variables x by y and vice versa, and also change all operations + by · and vice versa. So we obtain the Boole-De Morgan algebra T (A): x + y = x + y, x · y = x · y, x = y, y = x, (x + y)0 = x · y, (x · y)0 = x + y, x0 = y, y 0 = x. Note, that every lattice identity of the Boole-De Morgan algebra T (A) is equivalent to the certain hyperidentity (for hyperidentities see [16, 18–23]). For applications of mentioned binary superpositions see [12, 15–18]. Let B = (Q; {+, ·, 0 , 0, 1}) be a Boolean algebra. Denote its dual Boolean algebra by Bop , i.e., Bop = (Q; {·, +, 0 , 1, 0}). Consider the direct product B × Bop = (Q × Q; {∨, ∧, 0 , (0, 1), (1, 0)}) where (x1 , y1 ) ∨ (x2 , y2 ) = (x1 + x2 , y1 · y2 ), (x1 , y1 )∧(x2 , y2 ) = (x1 ·x2 , y1 +y2 ), (x1 , y1 )0 = (x01 , y10 ) for any x1 , x2 , y1 , y2 ∈ Q. 4

Defining one more unary operation ¯ on the set Q × Q by (x, y) = (y, x) we get the Boole-De Morgan algebra B × Bop . Now we prove a Stone-type representation theorem for Boole-De Morgan algebras. Theorem 2.2. Every Boole-De Morgan algebra is isomorphic to a subalgebra of the Boole-De Morgan algebra B × Bop for some Boolean algebra B. Proof: Suppose A = (Q; {+, ·,¯, 0 , 0, 1}) is a Boole-De Morgan algebra. From Stone’s representation theorem for Boolean algebras ( [8]) it follows that there exists a set I such that the Boolean algebra (Q; {+, ·, 0 , 0, 1}) is isomorphic to a subalgebra of the Boolean algebra (2I ; {∪, ∩, 0 , Ø, I}) = B where for a set X ⊆ I we define X 0 = I \ X. Let σ : Q → 2I be an embedding of the mentioned Boolean algebra in B. We define an embedding of the Boole-De Morgan algebra A in the Boole-De Morgan algebra B × Bop by the following rule: ϕ(s) = (σ(s), σ(s)), s ∈ Q. Indeed, for all s, t ∈ Q we have: ϕ(s + t) = (σ(s + t), σ(s + t)) = (σ(s + t), σ(s · t)) = (σ(s) ∪ σ(t), σ(s) ∩ σ(t)) = (σ(s), σ(s)) ∨ (σ(t), σ(t)) = ϕ(s) ∨ ϕ(t), ϕ(s · t) = (σ(s · t), σ(s · t)) = (σ(s · t), σ(s + t)) = (σ(s) ∩ σ(t), σ(s) ∪ σ(t)) = (σ(s), σ(s)) ∧ (σ(t), σ(t)) = ϕ(s) ∧ ϕ(t), ϕ(s) = (σ(s), σ(s)) = (σ(s), σ(s)) = ϕ(s), ϕ(s0 ) = (σ(s0 ), σ(s0 )) = (σ(s0 ), σ((s)0 )) = ((σ(s))0 , (σ(s))0 ) = (σ(s), σ(s))0 = (ϕ(s))0 . These equalities show that ϕ is a homomorphism. Obviously, ϕ is injective, hence it is an embedding.  In the next section we will establish a Birkhoff-type representation for BooleDe Morgan algebras. For a Boole-De Morgan algebra (Q; {+, ·,¯, 0 , 0, 1}) we define one more unary operation ∗ by the following way: x∗ = (x)0 = (x0 ). It is easy to see that (x + y)∗ = x∗ + y ∗ , (x · y)∗ = x∗ · y ∗ , x∗ = (x)∗ , (x∗ )0 = (x0 )∗ . Thus the mapping x → x∗ is an automorphism of the Boole-De Morgan algebra (Q; {+, ·,¯, 0 , 0, 1}). Also it is easy to see that x∗ = x if and only if x0 = x.

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3

Subdirectly irreducible Boole-De Morgan algebras

Recall that the zero congruence (denoted by O) is defined by the following rule: xOy ⇔ x = y. Definition 3.1. An algebra A is called subdirectly irreducible if any system of its nonzero congruence relations has a nonzero intersection, i.e., if there exist two elements u, v such that u 6= v and uαv for all nonzero congruences α of that algebra ( [3, 8]). Definition 3.2. Let the algebra A = (A; Σ) be a direct product of the algebras Ai = (Ai ; Σ) for i ∈ I with the projection maps πi : A → Ai for i ∈ I. A subalgebra B = (B; Σ) of A is called a subdirect product of the algebras Ai , for i ∈ I, if the projection map πi maps B onto Ai for all i ∈ I. Theorem 3.3 (Birkhoff’s subdirect representation theorem). In a variety V, every algebra can be represented as a subdirect product of subdirectly irreducible algebras in V ( [3, 8]). Following to [10,28,31], in this section we find all subdirectly irreducible BooleDe Morgan algebras up to isomorphism. But first we establish some preliminary results. Below everywhere A = (Q; {+, ·,¯, 0 , 0, 1}) is a Boole-De Morgan algebra. Lemma 3.4. For any element p ∈ Q we can define a congruence relation θp on algebra A by the following way: xθp y ⇔ x · p = y · p, x · p = y · p, x0 · p = y 0 · p, x∗ · p = y ∗ · p; x, y ∈ Q. Proof: Obviously θp is an equivalence relation. Let us prove that it is a congruence relation. Let xθp y and zθp t. We should prove that (x + z)θp (y + t), (x · z)θp (y · t), xθp y, x0 θp y 0 . The last two statements immediately follow from the definition of θp . Let us verify the first two statements. We have: (x + z) · p = x · p + z · p = y · p + t · p = (y + t) · p, x + z · p = x · z · p = (x · p) · (z · p) = (y · p) · (t · p) = y + t · p, (x + z)0 · p = x0 · z 0 · p = (x0 · p) · (z 0 · p) = (y 0 · p) · (t0 · p) = (y + t)0 · p, (x + z)∗ · p = (x∗ + z ∗ ) · p = (x∗ · p) + (z ∗ · p) = (y ∗ · p) + (t∗ · p) = (y + t)∗ · p. 6

These equalities prove that (x + z)θp (y + t). Now we have: (x · z) · p = (x · p) · (z · p) = (y · p) · (t · p) = (y · t) · p, x · z · p = (x + z) · p = (x · p) + (z · p) = (y · p) + (t · p) = y · t · p, (x · z)0 · p = (x0 + z 0 ) · p = (x0 · p) + (z 0 · p) = (y 0 · p) + (t0 · p) = (y · t)0 · p, (x · z)∗ · p = (x∗ · z ∗ ) · p = (x∗ · p) · (z ∗ · p) = (y ∗ · p) · (t∗ · p) = (y · t)∗ · p. These arguments prove that (x · z)θp (y · t).



Lemma 3.5. For any elements p, q ∈ Q the equality θp ∩ θq = θp+q is valid. Proof: Let xθp y and xθq y. Then we have: x · (p + q) = x · p + x · q = y · p + y · q = y · (p + q), x · (p + q) = x · p + x · q = y · p + y · q = y · (p + q), x0 · (p + q) = x0 · p + x0 · q = y 0 · p + y 0 · q = y 0 · (p + q), x∗ · (p + q) = x∗ · p + x∗ · q = y ∗ · p + y ∗ · q = y ∗ · (p + q). Hence xθp+q y. Let now xθp+q y. Then x · p = x · (p + q) · p = y · (p + q) · p = y · p, x · p = x · (p + q) · p = y · (p + q) · p = y · p, x0 · p = x0 · (p + q) · p = y 0 · (p + q) · p = y 0 · p, x∗ · p = x∗ · (p + q) · p = y ∗ · (p + q) · p = y ∗ · p. This shows that xθp y. Analogously we prove that xθq y.



Lemma 3.6. For any element p ∈ Q we have θp ∩ θp = O, where O is the zero congruence relation on algebra A. Proof: If xθp y and xθp y then we have: x = x · (x + p) = x · x · p = x · y · p = x · (y + p) = x · y + x · p = x · y + y · p = y · (x + p) = y · x · p = y · y · p = y · (y + p) = y. This proves the statement of the lemma.

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Lemma 3.7. For any element p ∈ Q the equality θp = O is true if and only if p ≥ p. Proof: If p ≥ p then O = θp ∩ θp = θp+p = θp . Now let θp = O. We claim that pθp (p + p). This follows from the following equalities: (p + p) · p = p · p, p + p · p = p · p, (p + p)0 · p = p0 · p∗ · p = 0 = p0 · p, (p + p)∗ · p = p∗ · p + p0 · p = p∗ · p. Now θp = O implies p = p + p, hence p ≥ p.



Lemma 3.8. Let A = (Q; {+, ·,¯, 0 , 0, 1}) be a subdirectly irreducible Boole-De Morgan algebra. Then for any elements x, y, z ∈ Q the following statements are true: (1) x and x are comparable, i.e., x ≤ x or x ≤ x, (2) if x > x and y > y, then x · y > x · y, (3) if y > y and z ≤ z, then y > z. Proof: (1) As A is subdirectly irreducible and θx ∩ θx = O, we have θx = O or θx = O. Now (1) follows immediately from Lemma 3.7. (2) Now let x > x and y > y, but x · y ≯ x · y. Then x · y ≤ x · y by (1). Lemma 3.7 implies θx ∩ θy = θx+y = θx·y = O. Hence θx = O or θy = O. Again using Lemma 3.7 we deduce that x ≥ x or y ≥ y, which contradicts the hypothesis. (3) Now let y > y and z ≤ z. If y · z > y · z, then from (2) we conclude that y · (y + z) = y · y · z > y · y · z = y + y · z = y · (y + z) ≥ y · (y + z), which is a contradiction. Hence y · z ≤ y · z (here we used (1)). So y ≥ y · z ≥ y · z = y + z ≥ z. But y 6= z, therefore y > z.  Lemma 3.9. Let A = (Q; {+, ·,¯, 0 , 0, 1}) be a subdirectly irreducible Boole-De Morgan algebra. Then we can define congruence relation ∼ on A by setting x ∼ y if and only if one of the following statements is true: (a) x > x and y > y, (b) x < x and y < y, (c) x = x = y = y. 8

Proof: Obviously ∼ is an equivalence relation. We prove that ∼ is a congruence relation. Let x ∼ y and z ∼ t. Obviously x ∼ y. Using the fact that u > u implies u0 < (u)0 = (u0 ) we conclude that x0 ∼ y 0 . Also note that u ≥ v implies u ≤ v. Now we shall prove that x + z ∼ y + t and x · z ∼ y · t. If x > x, y > y and z > z, t > t then using Lemma 3.8 we get: x + z ≥ x · z > x · z ≥ x + z, y + t ≥ y · t > y · t ≥ y + t. Therefore (x + z) ∼ (y + t) and (x · z) ∼ (y · t). If x > x, y > y and z < z, t < t then using Lemma 3.8 we conclude that x > z and y > t. Therefore x + z = x > x = x + z, y + t = y > y = y + t, x · z = z < z = x · z, y · t = t < t = y · t. Hence (x + z) ∼ (y + t) and (x · z) ∼ (y · t). The other cases are considered in an analogous way.  Lemma 3.10. Let A = (Q; {+, ·,¯, 0 , 0, 1}) be a subdirectly irreducible Boole-De Morgan algebra. Then for any element p ∈ Q we can define a congruence relation σp on A setting xσp y ⇔ x ∼ y and (x + x) · p = (y + y) · p and (x · x)0 · p = (y · y)0 · p. Proof: It is clear that σp is an equivalence relation. Let xσp y and zσp t. Obviously xσp y and x0 σp y 0 . Let us prove that (x + z)σp (y + t) and (x · z)σp (y · t). Suppose x > x, y > y and z > z, t > t. Then x + z ≥ x · z > x · z ≥ x + z and y + t ≥ y · t > y · t ≥ y + t. Also from xσp y and zσp t it follows that x · p = y · p, x∗ · p = y ∗ · p, z · p = t · p, z ∗ · p = t∗ · p. So (x + z + x + z) · p = (x + z) · p = (y + t) · p = (y + t + y + t) · p, ((x + z) · x + z))0 · p = (x + z)∗ · p = (y + t)∗ · p = ((y + t) · y + t))0 · p, (x · z + x · z) · p = (x · z) · p = (y · t) · p = (y · t + y · t) · p, ((x · z) · x · z))0 · p = (x · z)∗ · p = (y · t)∗ · p = ((y · t) · y · t))0 · p. These equalities show that (x + z)σp (y + t) and (x · z)σp (y · t). 9

Now let x > x, y > y and z ≤ z, t ≤ t. Then x > z and y > t. Also x · p = y · p, x∗ · p = y ∗ · p, z · p = t · p, z 0 · p = t0 · p. Therefore (x + z + x + z) · p = (x + x) · p = x · p = y · p = (y + y) · p = (y + t + y + t) · p, ((x + z) · x + z))0 · p = (x · x)0 · p = x∗ · p = y ∗ · p = (y · y)0 · p = ((y + t) · y + t))0 · p, (x · z + x · z) · p = (z + z) · p = z · p = t · p = (t + t) · p = (y · t + y · t) · p, ((x · z) · x · z))0 · p = (z · z)0 · p = z 0 · p = t0 · p = (t · t)0 · p = ((y · t) · y · t))0 · p. Again we get (x + z)σp (y + t) and (x · z)σp (y · t). The other cases are considered analogously.  Lemma 3.11. Let A = (Q; {+, ·,¯, 0 , 0, 1}) be a subdirectly irreducible Boole-De Morgan algebra. Then for any element p ∈ Q we have σp = O or σp = O. Proof: Let xσp y and xσp y. Without loss of generality we can suppose that x ≥ x, y ≥ y. Then x · p = y · p, x∗ · p = y ∗ · p, x · p = y · p, x∗ · p = y ∗ · p. Hence x · p0 = (x∗ · p)∗ = (y ∗ · p)∗ = y · p0 and x = x · (p + p0 ) = x · p + x · p0 = y · p + y · p0 = y · (p + p0 ) = y. We conclude that σp ∩ σp = O. Now as A is subdirectly irreducible, σp = O or σp = O.  Theorem 3.12. The Boole-De Morgan algebras BM 2 and BM 4 are subdirectly irreducible, and those algebras are the only nontrivial subdirectly irreducible BooleDe Morgan algebras up to isomorphism. Proof: The algebras BM 2 and BM 4 are clearly subdirectly irreducible. Now let A = (Q; {+, ·,¯, 0 , 0, 1}) be a subdirectly irreducible Boole-De Morgan algebra. Let u ∈ Q be an element with u > u and u 6= 1. Denote v = u + u0 . We claim that v = u · u∗ < v. Indeed, if v ≤ v then u0 ≤ u. Hence 1 = u + u0 = u, which is a contradiction. Thus v > v, so u ∼ v. Also v ∗ = v. We claim that 1σv u and 1σv v. It follows from the following statements: (u + u) · v = v = (1 + 1) · v, (u · u)0 · v = u∗ · v = v = (1 · 1)0 · v, (v + v) · v = v = (1 + 1) · v, (v · v)0 · v = v ∗ · v = v = (1 · 1)0 · v.

10

Now from Lemma 3.11 it follows that 1 = u or 1 = v. As the first equality is not true, the second must be valid, i.e., u + u0 = 1. Hence u = u · (u + u0 ) = u · u = u. But this is a contradiction. Thus we have x = x for all x ∈ Q \ {0, 1}. Now let x, y ∈ Q \ {0, 1} and x 6= y. If x + y 6= 1 then x · y = x + y = x + y and therefore x = y. This contradicts our assumption, so x + y = 1. Similarly for any x, y ∈ Q \ {0, 1} the equality x · y = 0 is true. Now if there were three distinct elements u, v, w ∈ Q \ {0, 1} then u = u · (v + w) = u · v + u · w = 0 + 0 = 0. We arrived at a contradiction, therefore there are at most two elements different from 0 and 1, i.e., |Q| ≤ 4. But |Q| = 6 3 as (Q; {+, ·, 0 , 0, 1}) is a Boolean algebra. If |Q| = 2 then A is isomorphic to BM 2 ; if |Q| = 4 then A is isomorphic to BM 4 . This completes the proof.  Now from the Birkhoff’s subdirect representation theorem we conclude the following result. Corollary 3.13. Every Boole-De Morgan algebra is isomorphic to a subdirect product of algebras BM 2 and BM 4 . As BM 2 is a subalgebra of BM 4 , we conclude that every Boole-De Morgan algebra is isomorphic to a subalgebra of a direct power of the algebra BM 4 . For a variety V of algebras denote by Eq(V) the set of all identities that are satisfied in all algebras of V (the equational theory of the variety V). Also for M ∈ V by Eq(M) denote the set of all identities of that algebra (the equational theory of the algebra M). Also by BM denote the variety of all Boole-De Morgan algebras. Corollary 3.14. Eq(BM) = Eq(BM 4 ). Proof: Obviously Eq(BM) ⊆ Eq(BM 4 ). Now if M ∈ V then M is isomorphic to a subalgebra of direct power of algebra BM 4 . But any identity of the algebra BM 4 is valid also in a direct power of that algebra, and hence, in M. Therefore Eq(BM) ⊇ Eq(BM 4 ).  Corollary 3.15. The freen n-generated Boole-De Morgan algebra is isomorphic to BM the subalgebra of BM 4 4 generated by the projections. Proof: For any element (term) ω of the free n-generated Boole-De Morgan algebra we correspond the term operation (function) ω ˆ : Dn → D. Corollary 3.14 implies that this mapping ω → ω ˆ is injective. The homomorphism of the correspondence follows immediately from the definitions of the operations.  Corollary 3.16. Every finitely generated free Boole-De Morgan algebra is finite. Corollary 3.17. Every finitely generated Boole-De Morgan algebra is finite. 11

4

Quasi-De Morgan functions

Recall that D = {0, a, b, 1} and B = {0, 1}. Let us construct a one-to-one correspondence between the sets D and B × B as follows: 0 ↔ (0, 0), a ↔ (1, 0), b ↔ (0, 1), 1 ↔ (1, 1). We define the operations +, ·,¯,

0

on the set B × B as follows:

(u, v) = (v 0 , u0 ), (u, v)0 = (u0 , v 0 ), (u1 , v1 ) + (u2 , v2 ) = (u1 + u2 , v1 + v2 ), (u1 , v1 ) · (u2 , v2 ) = (u1 · u2 , v1 · v2 ) (here the operations on the right hand side are the operations of the Boole-De Morgan algebra BM 2 ). We get the Boole-De Morgan algebra (B×B; {+, ·,¯, 0 , 0, 1}) = B × B (see Section 2), which is isomorphic to the algebra BM 4 (the one-to-one correspondence described above is an isomorphism). However, if the ordered pair (y, z) ∈ B × B corresponds to x ∈ D then we will write x = (y, z) (this causes no confusion). For x ∈ D let   x, if x = 0, 1, a, if x = b, x∗ =  b, if x = a. ∗ The unary operation can also be defined on B × B taking into account the isomorphism described above. In result we get (u, v)∗ = (v, u) = (v ∗ , u∗ ), u, v ∈ B. It is clear that x∗ = (x)0 = x0 (which agrees with the notation from the previous section). The following two concepts of quasi-De Morgan function and De Morgan function are introduced in [24]. Definition 4.1. A function f : Dn → D is called a quasi-De Morgan function if the following conditions hold: (1) if xi ∈ {0, 1}, i = 1, . . . , n, then f (x1 , . . . , xn ) ∈ {0, 1}, (2) if xi ∈ D, i = 1, . . . , n, then f (x∗1 , . . . , x∗n ) = (f (x1 , . . . , xn ))∗ , In terms of clone theory, Condition (1) means that the function f preserves the unary relation {0, 1} ⊆ D, and Condition (2) means that f preserves the binary relation {(0, 0), (a, b), (b, a), (1, 1)} ⊆ D2 , which is the graph of the automorphism x 7→ x∗ . For c = (c1 , . . . , cn ), d = (d1 , . . . , dn ) ∈ Dn we say that d is a permitted modification of c if for some k (1 ≤ k ≤ n) we have di = ci for all 1 ≤ i ≤ n, i 6= k, and  a, if ck = 0, dk = 1, if ck = b. 12

Definition 4.2. A quasi-De Morgan function f : Dn → D is called a De Morgan function if it satisfies the following condition: (3) if x, y ∈ Dn with f (x) 6= b and y is a permitted modification of x then f (y) ∈ {f (x), a}. In terms of clone theory the Condition (3) means that f preserves the order relation ρ = {(b, b), (b, 0), (b, 1), (b, a), (0, 0), (0, a), (1, 1), (1, a), (a, a)} ⊆ D2 . Notice that Condition (1) is a consequence of Condition (2), but however it is more convenient to write it as a separate condition. Note that it follows from Condition (1) that every quasi-De Morgan function is an extension of some Boolean function. Notice that the constant functions f = 1 and f = 0 are quasi-De Morgan functions, but the constant functions f = a and f = b are not. This means that 0 and 1 are the only constant quasi-De Morgan functions. Further examples of quasi-De Morgan functions are f (x) = x, g(x) = x, h(x, y) = x · y, q(x, y) = x + y, p(x) = x0 , where the operations on the right hand side are the operations of the Boole-De Morgan algebra BM 4 . Also note that the function p is an example of quasi-De Morgan function which is not a De Morgan function. Below, for xi ∈ D we denote by (yi , zi ) the pair from B × B which corresponds to xi , i.e., xi = (yi , zi ). Theorem 4.3. A function f : Dn → D is a quasi-De Morgan function if and only if there exists a Boolean function ϕ : B 2n → B such that f (x1 , . . . , xn ) = (ϕ(y1 , . . . , yn , z10 , . . . , zn0 ), ϕ(z1 , . . . , zn , y10 , . . . , yn0 )),

(4.1)

for all x1 , . . . , xn ∈ D. Proof: Let f satisfy the condition (4.1) for some Boolean function ϕ. If xi ∈ {0, 1}, then yi = zi and ϕ(y1 , . . . , yn , z10 , . . . , zn0 ) = ϕ(z1 , . . . , zn , y10 , . . . , yn0 ). Hence, f (x1 , . . . , xn ) ∈ B. Thus, the Condition (1) holds for f . Now let us check the Condition (2). To do this recall that (u, v)∗ = (v, u). Hence, f (x∗1 , . . . , x∗n ) = (ϕ(z1 , . . . , zn , y10 , . . . , yn0 ), ϕ(y1 , . . . , yn , z10 , . . . , zn0 )) = (ϕ(y1 , . . . , yn , z10 , . . . , zn0 ), ϕ(z1 , . . . , zn , y10 , . . . , yn0 ))∗ = (f (x1 , . . . , xn ))∗ . Now suppose f is a quasi-De Morgan function, and let us prove that there exists a Boolean function ϕ with condition (4.1). First we prove that there are n at most 24 functions for which Conditions (1) and (2) hold. To see this notice that there are 2n n-tuples (u1 , . . . , un ) ∈ B n . For such n-tuples f can take only two values (by Condition (1)). Further, if the n-tuple (v1 , . . . , vn ) ∈ Dn contains a 13

or b, then (v1∗ , . . . , vn∗ ) 6= (v1 , . . . , vn ) and f (v1∗ , . . . , vn∗ ) is uniquely determined by f (v1 , . . . , vn ) (by Condition (2)). There are 4n − 2n such n-tuples. Thus, the num4n −2n n n n n n ber of quasi-De Morgan functions does not exceed 22 · 4 2 = 22 · 24 −2 = 24 . It is clear that for a function f with (4.1) there exists exactly one Boolean 2n n function ϕ that satisfies (4.1). Therefore, there are 22 = 24 such functions. And all those functions satisfy Conditions (1) and (2). Hence, all functions f : Dn → D satisfying (1) and (2) also satisfy (4.1) for some Boolean function ϕ.  As we mentioned in the proof, for a quasi-De Morgan function f : Dn → D there exists a unique Boolean function ϕ : B 2n → B which satisfies (4.1). To emphasize that ϕ is the unique Boolean function corresponding to f , we denote it by ϕf . n

Corollary 4.4. There are exactly 24 quasi-De Morgan functions of n variables. Denote the set of all quasi-De Morgan functions of n variables by BMn . For the functions f, g : Dn → D define f + g, f · g, f and f 0 by the standard way, i.e., (f + g)(x) = f (x) + g(x), (f · g)(x) = f (x) · g(x), f (x) = f (x), f 0 (x) = (f (x))0 , x ∈ Dn , where the operations on the right hand side are the operations of the Boole-De Morgan algebra BM 4 . Theorem 4.5. The set BMn is closed under operations +, ·,¯, 0 , i.e., if f, g ∈ BMn , then f + g, f · g, f , f 0 ∈ BMn . Proof: We will use the facts that (x + y)∗ = x∗ + y ∗ , (x · y)∗ = x∗ · y ∗ , x∗ = (x)∗ , (x∗ )0 = (x0 )∗ . We have: (f + g)(x∗1 , . . . , x∗n ) = f (x∗1 , . . . , x∗n ) + g(x∗1 , . . . , x∗n ) = (f (x1 , . . . , xn ))∗ + (g(x1 , . . . , xn ))∗ = ((f + g)(x1 , . . . , xn ))∗ , (f · g)(x∗1 , . . . , x∗n ) = f (x∗1 , . . . , x∗n ) · g(x∗1 , . . . , x∗n ) = (f (x1 , . . . , xn ))∗ · (g(x1 , . . . , xn ))∗ = ((f · g)(x1 , . . . , xn ))∗ , f (x∗1 , . . . , x∗n ) = f (x∗1 , . . . , x∗n ) = (f (x1 , . . . , xn ))∗ = ∗

∗

(f (x1 , . . . , xn )) = (f (x1 , . . . , xn )) , 0

f 0 (x∗1 , . . . , x∗n ) = (f (x∗1 , . . . , x∗n ))0 = ((f (x1 , . . . , xn ))∗ ) = ∗

∗

((f (x1 , . . . , xn ))0 ) = (f 0 (x1 , . . . , xn )) . These equalities prove the statement of the theorem.

14



Thus, we get an algebra: BMn = (BMn , {+, ·,¯ , 0 , 0, 1}) (here 0 and 1 are the constant quasi-De Morgan functions), which obviously is a Boole-De Morgan algebra. For a set S ⊆ 2{1,...,n} × 2{1,...,n} define the function fS : Dn → D by the following way: ! Y Y X Y Y x0i · xi · x∗i , (4.2) fS (x1 , . . . , xn ) = xi · (s1 ,s2 )∈S

i∈s1

i∈s1

i∈s2

i∈s2

where the operations on the right hand side are the operations of BM 4 (cf. [21– 23]). Notice that fS does not depend on the order of the elements in the set S. Also we set fØ = 0 and f{(Ø,Ø)} = 1. Let us consider the projection functions δni (x1 , . . . , xn ) = xi , i = 1, . . . , n, as functions Dn → D. Obviously, δni is a quasi-De Morgan function for each i. According to (4.2), for any set S ⊆ 2{1,...,n} × 2{1,...,n} we have: ! Y X Y Y Y 0 ∗ (δni ) · (δni ) . δni · fS = δni · (s1 ,s2 )∈S

i∈s1

i∈s1

i∈s2

i∈s2

Hence, fS ∈ BMn , i.e., fS is a quasi-De Morgan function for any set S ⊆ 2{1,...,n} × 2{1,...,n} . For s = (s1 , s2 ) ∈ 2{1,...,n} × 2{1,...,n} let s0 = s1 ∪ {n + i : i ∈ s2 } ∈ 2{1,...,2n} , and for S ⊆ 2{1,...,n} × 2{1,...,n} let S 0 = {s0 : s ∈ S} ⊆ 2{1,...,2n} . In this
way we give a one-to-one correspondence between the sets P 2{1,...,n} × 2{1,...,n} and
P 2{1,...,2n} . Now, for any quasi-De Morgan function f ∈ BMn from Theorem 4.3 we conclude that there exists a set S 0 ⊆ 2{1,...,2n} such that:

15

f (x1 , . . . , xn ) = (ϕf (y1 , . . . , yn , z10 , . . . , zn0 ), ϕf (z1 , . . . , zn , y10 , . . . , yn0 )) =   X Y Y  y · yi0 · i 

s0 ∈S 0

i∈s0 1≤i≤n

i∈s0 1≤i≤n

Y

Y

0 zi−n ·

i∈s0 n+1≤i≤2n

i∈s0 n+1≤i≤2n



 zi−n  ,  !

Y

X Y Y  z · zi0 · i 

s0 ∈S 0

i∈s0

1≤i≤n

i∈s0 1≤i≤n

Y

0 yi−n ·

i∈s0

i∈s0 n+1≤i≤2n

n+1≤i≤2n

 yi−n  

=





X Y Y  (y , z ) · (yi , zi )0 · i i 

s0 ∈S 0

i∈s0 1≤i≤n

i∈s0 1≤i≤n

Y

Y

(yi−n , zi−n ) ·

i∈s0 n+1≤i≤2n

i∈s0 n+1≤i≤2n

 (yi−n , zi−n )∗  =

! X

Y

(s1 ,s2 )∈S

i∈s1

xi ·

Y i∈s1

x0i ·

Y

xi ·

i∈s2

Y

x∗i

= fS (x1 , . . . , xn ),

i∈s2

where S is the subset of 2{1,...,n} × 2{1,...,n} corresponding to S 0 . Moreover, the number of all quasi-De Morgan functions of n variables is the same as the number of all subsets of 2{1,...,n} × 2{1,...,n} . Hence, we get the following result. Theorem 4.6. For any quasi-De Morgan function f of n variables there exists a unique set S ⊆ 2{1,...,n} × 2{1,...,n} such that f = fS . In particular, fS1 6= fS2 if S1 6= S2 . Thus every quasi-De Morgan function can be uniquely presented in the form (4.2). This form is called the disjunctive normal form (or briefly - DNF) of quasiDe Morgan function f . Notice that from Theorems 4.3 and 4.6 we get an algorithm which, given a quasi-De Morgan function, gives its disjunctive normal form. We can also prove that every quasi-De Morgan function can be uniquely presented in conjunctive normal form (CNF), i.e., in the following form: ! X X Y X X x0i + xi + x∗i . xi + (s1 ,s2 )∈S

i∈s1

i∈s1

i∈s2

i∈s2

Below we will use the concept of essential variable (and essential dependence) of quasi-De Morgan functions. The definitions are the same as in case of Boolean functions and so we do not give them here. 16

Using arguments similar to those given above we can prove the following theorem. Theorem 4.7. For a quasi-De Morgan function f the corresponding Boolean function ϕf does not essentially depend on the last n variables if and only if f can be represented as a term function with functional symbols +, ·, 0 , i.e., f is a term function of the Boolean algebra (D; {+, ·, 0 , 0, 1}). Now let us prove one of the main results of our paper. Theorem 4.8. (Functional representation theorem) The algebra BMn is a free Boole-De Morgan algebra with the system of free generators ∆ = {δn1 , . . . , δnn }. Hence every free n-generated Boole-De Morgan algebra is isomorphic to the Boole-De Morgan algebra BMn . Proof: Let F = (Q; {+, ·,¯, 0 , 0, 1}) be a Boole-De Morgan algebra and µ : ∆ → Q be a mapping. We prove that there exists a unique homomorphism ν : BMn → F with ν|∆ = µ. According to the Theorem 4.6, any element f ∈ BMn can be presented in the form ! Y Y X Y Y 0 ∗ (δni ) · δni · (δni ) f= δni · (s1 ,s2 )∈S

i∈s1

i∈s2

i∈s1

i∈s2

for the uniquely determined set S ⊆ 2{1,...,n} × 2{1,...,n} . Set ! ν(f ) =

X

Y

(s1 ,s2 )∈S

i∈s1

µ(δni ) ·

Y

0

(µ(δni )) ·

Y i∈s2

i∈s1

µ(δni ) ·

Y

∗

(µ(δni ))

.

i∈s2

Obviously, ν(δni ) = µ(δni ), i = 1, . . . , n. Let us prove that ν is a homomorphism. Let f1 , f2 ∈ BMn . Then f1 = fS1 , f2 = fS2 for some sets S1 , S2 ⊆ 2{1,...,n} × 2{1,...,n} . Let the terms w1 and w2 be the disjunctive normal forms of quasi-De Morgan functions f1 and f2 respectively; and let w be the disjunctive normal form of quasi-De Morgan function f1 + f2 . Then the identity w = w1 + w2 holds in the Boole-De Morgan algebra BM 4 . From Corollary 3.14 it follows that the identity w = w1 + w2 holds in the Boole-De Morgan algebra F = (Q; {+, ·,¯ , 0 , 0, 1}). Hence ν(f1 + f2 ) = ν(f1 ) + ν(f2 ). Analogously we prove that ν(f1 · f2 ) = ν(f1 ) · ν(f2 ), ν(f ) = ν(f ), ν(f 0 ) = (ν(f ))0 . Obviously ν(0) = 0, ν(1) = 1. Therefore ν is a homomorphism. Uniqueness of ν is evident.  Now let us prove the following characterization of quasi-De Morgan functions. 17

Theorem 4.9. Quasi-De Morgan functions are precisely the term functions of the Boole-De Morgan algebra BM 4 , i.e., the set of all quasi-De Morgan functions is the clone of term functions of the Boole-De Morgan algebra BM 4 (for clones see [18, 29]). Proof: As every quasi-De Morgan function can be represented in DNF, we conclude that quasi-De Morgan functions are term functions of the algebra BM 4 . We shall prove the converse, i.e., we shall prove that any term function of the algebra BM 4 is a quasi-De Morgan function. We induct on the structure of terms. (Again for a term ω we will denote by ω b the term function of the algebra BM 4 corresponding to that term.) If ω = xi is a variable, then the corresponding term function ω b is a projection function, which is a quasi-De Morgan function. If ω = 0 or ω = 1 then ω b is either the constant quasi-De Morgan function 0 or 1. Now suppose ω = ω1 +ω2 or ω = ω1 ·ω2 , where ω b1 and ω b2 are quasi-De Morgan functions. In the first case ω b = ω\ b1 + ω b2 , in the second case ω b = ω\ b1 · ω b2 . And 1 + ω2 = ω 1 · ω2 = ω as we know the sum and product of two quasi-De Morgan functions are quasi-De Morgan functions too (see Theorem 4.5). Hence ω b is a quasi-De Morgan function. 0 b Then in Finally, suppose ω = λ or ω = λ , and λ is a quasi-De Morgan 0  function. b d 0) = λ b . Both of these b and in the second case ω b = (λ the first case ω b =λ=λ functions are quasi-De Morgan functions due to Theorem 4.5. This finishes the proof. 

5

Representation of quasi-De Morgan functions by polynomials

In this section we will give some representations of quasi-De Morgan functions by polynomials analogous to Zhegalkin polynomials (or polynomials modulo 2) of Boolean functions. First recall that in the theory of Boolean functions the exclusive-or function is defined as follows: x ⊕ y = xy 0 + x0 y. This is a binary operation on the set B and the algebra (B; {⊕, ·}) is a field (it is the two-element Galois field GF(2) = Z2 ). Recall the following well known result. Theorem 5.1. ( [7]) For every Boolean function f : B n → B there exists a unique mapping α : P({1, . . . , n}) → {0, 1} such that M Y f (x1 , . . . , xn ) = α(A) · xi , (5.1) A∈P({1,...,n})

18

i∈A

where the operations on the right hand side are the operations of the two-element field GF(2) = Z2 . The right hand side of (5.1) is a polynomial modulo 2. Those polynomials are called also Zhegalkin polynomials for Boolean functions [32]. We will try to generalize the operation ⊕ and define an analogous operation on the set D which satisfies some properties of the operation ⊕ on the set B. We will designate the operation on D by the same symbol ⊕. As we want to consider polynomials with respect to operations ⊕ and ·, it will be natural to require the operation ⊕ to be associative and commutative and also distributive with respect to ·. As we consider quasi-De Morgan functions we also require that x ⊕ y be a quasi-De Morgan function. Also we want this operation to be an extension of the analogous operation on B, i.e., the restriction of the function x ⊕ y on B be the exclusive-or function defined above. Let us recall the one-to-one correspondence between the sets D and B × B described in Section 3: 0 ↔ (0, 0), a ↔ (1, 0), b ↔ (0, 1), 1 ↔ (1, 1). Now we can define an operation ⊕ on B × B componentwisely (i.e., (u1 , v1 ) ⊕ (u2 , v2 ) = (u1 ⊕ u2 , v1 ⊕ v2 )) and taking into account the above one-to-one correspondence we can define the operation ⊕ on D. We get the following table of that operation: ⊕ 0 1 a b

0 0 1 a b

1 1 0 b a

a a b 0 1

b b a 1 0

The algebras (D; {⊕, ·}) and (B × B; {⊕, ·}) are isomorphic. Hence the operation ⊕ on D satisfies the required conditions. Also notice that x ⊕ y = x0 y + xy 0 , where the operations on the right hand side are the operations of the Boole-De Morgan algebra BM 4 . Now we can define Zhegalkin polynomials of first kind on the set D. Those are the polynomials of the form (5.1), only the domain of variables now is the four-element set D, and ⊕ is the operation on the set D defined above and · is the multiplication of the Boole-De Morgan algebra BM 4 . The functions x ⊕ y, x · y and constant functions 0, 1 are quasi-De Morgan functions, hence all Zhegalkin polynomials of first kind considered as functions from Dn into D are quasi-De Morgan functions. Here we restrict the coefficients to be only 0 and 1 (and not a 19

or b) because the constant functions a and b are not quasi-De Morgan functions and so if we let the coefficients to be a or b then the polynomial could be a non quasi-De Morgan function. Two Zhegalkin polynomials of first kind are called equal if the corresponding coefficients are equal. Theorem 5.2. A quasi-De Morgan function f : Dn → D can be represented as a Zhegalkin polynomial of first kind if and only if the corresponding Boolean function ϕf (t1 , . . . , t2n ) does not essentially depend on the variables tn+1 , . . . , t2n . If this is the case then the representation is unique. Proof: Let ϕf (t1 , . . . , t2n ) do not essentially depend on the last n variables. Thus we can assume that ϕf = ϕf (t1 , . . . , tn ) is a Boolean function of n variables. From Theorem 5.1 it follows that there exists a unique mapping α : P({1, . . . , n}) → {0, 1} such that M Y xi . (5.2) ϕf (x1 , . . . , xn ) = α(A) · A∈P({1,...,n})

i∈A

Therefore f (x1 , . . . , xn ) = (ϕf (y1 , . . . , yn ), ϕf (z1 , . . . , zn )) =   M Y M Y  α(A) · yi , α(A) · zi  = A∈P({1,...,n})

M A∈P({1,...,n})

i∈A

A∈P({1,...,n})

Y α(A) · (yi , zi ) = i∈A

M A∈P({1,...,n})

i∈A

α(A) ·

Y

xi .

i∈A

Thus the “if”-part of the theorem is proved. Let us prove the “only if”-part. n There are exactly 22 functions for which ϕf (t1 , . . . , t2n ) does not essenn tially depend on the variables tn+1 , . . . , t2n , and also there are exactly 22 Zhegalkin polynomials of first kind, therefore only such functions can be represented as Zhegalkin polynomials of first kind and the representation is unique.  This theorem shows that two Zhegalkin polynomials of first kind are equal (as polynomials) if and only if they are equal as functions, i.e., their values are equal for any values of variables. It is easy to verify that for a quasi-De Morgan function f the corresponding Boolean function ϕf does not essentially depend on the last n variables if and only if f preserves the binary relation σ = {(0, 0), (0, b), (b, 0), (b, b), (a, a), (a, 1), (1, 1), (1, a)} ⊆ D2 . Thus a quasi-De Morgan function is a Zhegalkin polynomial of first kind if and only if it preserves the binary relation σ. 20

From Theorem 4.9 we deduce that the term functions of the four-element Boolean algebra (D; {+, ·, 0 , 0, 1}) and only they can be represented as Zhegalkin polynomials of first kind. This result also follows from the following equalities: u0 = 1⊕u, u ∈ D and u+v = (u0 ·v 0 )0 = 1⊕(1⊕u)·(1⊕v) = u⊕v ⊕u·v, u, v ∈ D. These equalities give us an algorithm which, given a quasi-De Morgan function, gives its representation as a Zhegalkin polynomial of first kind. Example 5.3. Let us represent the quasi-De Morgan function f (x, y, z) = xy 0 +yz 0 as a Zhegalkin polynomial of first kind. Taking into account the equality u ⊕ u = 0, u ∈ D, we get: f (x, y, z) = ((xy 0 )0 · (yz 0 )0 )0 = 1 ⊕ (1 ⊕ x(1 ⊕ y))(1 ⊕ y(1 ⊕ z)) = 1 ⊕ 1 ⊕ y ⊕ yz ⊕ x ⊕ xy ⊕ xyz ⊕ xy ⊕ xy ⊕ xyz = x ⊕ y ⊕ xy ⊕ yz. A quasi-De Morgan function f is called monotone if it preserves the order of the lattice (D; {+, ·}). Theorem 5.4. ( [27]) A De Morgan function f (x1 , . . . , xn ) is monotone if and only if the corresponding Boolean function ϕf (t1 , . . . , t2n ) does not essentially depend on the variables tn+1 , . . . , t2n . Now we deduce the following result. Corollary 5.5. A De Morgan function can be represented as a Zhegalkin polynomial of first kind if and only if it is monotone. Now we define Zhegalkin polynomials of second kind. It is easy to see that the algebra (D; {⊕, ·}) is a Boolean ring isomorphic to the ring Z2 × Z2 (here · is the multiplication of the algebra BM 4 ). Now we define binary operation ⊗ on D such that the algebra (D; {⊕, ⊗}) is the four-element Galois field GF(4) (as it is known, there is only one finite field with the given number of elements up to isomorphism). The operation ⊕ is defined above. We give the table of ⊗ below: ⊗ 0 1 a b

0 1 0 0 0 1 0 a 0 b

21

a 0 a b 1

b 0 b 1 a

It is clear that the function x ⊗ y is a quasi-De Morgan function. Denote I = {0, 1, 2, 3}. Now we can consider polynomials of the following form: Pα (x1 , . . . , xn ) =

M (k1 ,...,kn

α(k1 , . . . , kn ) ⊗

)∈I n

n O

xki i ,

(5.3)

i=1

where α : I n → {0, 1} and   x . . ⊗ x}, if 1 ≤ k ≤ 3, | ⊗ .{z xk = k  1, if k = 0. Those polynomials are also generalizations of Zhegalkin polynomials for Boolean functions, i.e., if we consider them as functions from Dn into D then their restrictions on B are precisely Zhegalkin polynomials for Boolean functions. We call those polynomials Zhegalkin polynomials of second kind. Two such polynomials Pα1 and Pα2 are called equal if α1 = α2 . n Clearly the number of such polynomials is 24 . So if we consider them as functions n then their number will not exceed 24 . But we claim that their number (as funcn tions) is exactly 24 . This means that for different mappings α : I n → {0, 1} the corresponding Zhegalkin polynomials of second kind are not equal (as functions). It is well known (from the Lagrange’s interpolation theorem for functions of several variables) that if F = (F ; {⊕, ⊗}) is a finite field then all functions from F n into F are polynomials. But here the coefficients of a polynomial can be any elements of F . Thus if we consider functions α : I n → D instead of functions α : I n → {0, 1}, then all functions from Dn into D will be polynomials of the form n n (5.3). As there are 44 such polynomials and 44 functions from Dn into D, we conclude that different polynomials are different as functions. Further, x ⊕ y and x ⊗ y are quasi-De Morgan functions, hence all Zhegalkin n polynomials of second kind are quasi-De Morgan functions as well. There are 24 n quasi-De Morgan functions of n variables and 24 Zhegalkin polynomials of second kind, therefore all quasi-De Morgan functions are Zhegalkin polynomials of second kind. Thus, we proved the following result. Theorem 5.6. For every quasi-De Morgan function f : Dn → D there exists a unique mapping α : I n → {0, 1} such that f (x1 , . . . , xn ) = Pα (x1 , . . . , xn ), i.e., every quasi-De Morgan function can be uniquely represented as a Zhegalkin polynomial of second kind.

22

Now notice that x0 = 1 ⊕ x, x = 1 ⊕ (x ⊗ x) = 1 ⊕ x2 , x∗ = x ⊗ x = x2 , x · y = (x2 ⊗ y 2 ) ⊕ (x2 ⊗ y) ⊕ (x ⊗ y 2 ), x + y = (x0 · y 0 )0 = 1 ⊕ (1 ⊕ x) · (1 ⊕ y) = x ⊕ y ⊕ x · y = x ⊕ y ⊕ (x2 ⊗ y 2 ) ⊕ (x2 ⊗ y) ⊕ (x ⊗ y 2 ). These equalities give another proof of Theorem 5.6. Indeed, replacing the terms of the form u0 , u, u∗ , u + v, u · v in the DNF of f by corresponding polynomials according to the above equalities, we get the desired polynomial. The uniqueness of representation follows from the fact that the number of quasi-De Morgan functions of n variables is equal to the number of Zhegalkin polynomials of second kind of n variables. Example 5.7. Let us find the representation of the quasi-De Morgan function g(x, y) = x0 · y ∗ as a Zhegalkin polynomial of second kind. We have: x0 · y ∗ = (1 ⊕ x) · y 2 = ((1 ⊕ x)2 ⊗ y 4 ) ⊕ ((1 ⊕ x)2 ⊗ y 2 ) ⊕ ((1 ⊕ x) ⊗ y 4 ) = ((1 ⊕ x2 ) ⊗ y) ⊕ ((1 ⊕ x2 ) ⊗ y 2 ) ⊕ ((1 ⊕ x) ⊗ y) = (y ⊕ (x2 ⊗ y)) ⊕ (y 2 ⊕ (x2 ⊗ y 2 )) ⊕ (y ⊕ (x ⊗ y)) = (x2 ⊗ y 2 ) ⊕ (x2 ⊗ y) ⊕ (x ⊗ y) ⊕ y 2 . Here we used the identity y 4 = y which is true since (D; {⊕, ⊗}) is the four-element field. Acknowledgement Thanks to the referee for the useful remarks. This research was supported by State Committee Science of Republic of Armenia.

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