A FUNCTIONAL APPROACH TO d-ALGEBRAS

Abstract

In this paper we discuss a functional approach to obtain a lattice-like structure in d-algebras, and obtain an exact analog of De Morgan law and some other properties.

1. INTRODUCTION

Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras ([8, 9]). BCK-algebras have some connections with other areas: D. Mundici [13] proved that MV -algebras are categorically equivalent to bounded commutative BCK-algebras, and J. Meng [11] proved that implicative commutative semigroups are equivalent to a class of BCK-algebras. It is well known that bounded commutative BCK-algebras, D-posets and MV -algebras are logically equivalent each other (see [4, p. 420]). We refer useful textbooks for BCK/BCI-algebra to [4, 6, 7, 12, 17]. J. Neggers and H. S. Kim ([14]) introduced the notion of d-algebras which is a useful generalization of BCK-algebras, and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs. J. S. Han et al. ([5]) defined a variety of special d-algebras, such as strong d-algebras, (weakly) selective d-algebras and others. The main assertion is that the squared algebra (X; □, 0) of a d-algebra is a d-algebra if and only if the root (X; ∗, 0) of the squared algebra (X; □, 0) is a strong d-algebra. Recently, the present author with H. S. Kim and J. Neggers ([10]) explored properties of the set of d-units of a d-algebra. It was noted that many d-algebras are weakly associative, and the existence of non-weakly associative d/BCK-algebras was demonstrated. Moreover, they discussed the notions of a d-integral domain and a left-injectivity in d/BCK-algebras. We refer to [1, 2, 15, 16] for more information on d-algebras.

In this paper we discuss a functional approach to obtain a lattice-like structure in d-algebras, and obtain an exact analog of De Morgan law and some other properties.

 

2. PRELIMINARIES

An (ordinary) d-algebra ([14]) is a non-empty set X with a constant 0 and a binary operation “ ∗ ” satisfying the following axioms:

(D1) x ∗ x = 0, (D2) 0 ∗ x = 0, (D3) x ∗ y = 0 and y ∗ x = 0 imply x = y for all x, y ∈ X.

A BCK-algebra is a d-algebra X satisfying the following additional axioms:

(D4) (x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (D5) (x ∗ (x ∗ y)) ∗ y = 0 for all x, y, z ∈ X.

Example 2.1 ([14]). Consider the real numbers R, and suppose that (R; ∗, e) has the multiplication

x ∗ y = (x − y)(x − e) + e

Then x∗x = e; e ∗x = e; x∗ y = y ∗x = e yields (x−y)(x−e) = 0, (y −x)(y −e) = e and x = y or x = e = y, i.e., x = y, i.e., (R; ∗, e) is a d-algebra.

 

3. A FUNCTIONAL APPROACH TO d-ALGEBRAS

Let (X, ∗, 0) be a d-algebra. A map ϕ : X → X is said to be order reversing if x ∗ y = 0 then ϕ(y) ∗ ϕ(x) = 0 for all x, y ∈ X; self-inverse if ϕ(ϕ(x)) = x for all x ∈ X; an anti-homomorphism if ϕ(x ∗ y) = ϕ(y) ∗ ϕ(x) = 0 for all x, y ∈ X; a homomorphims if ϕ(x ∗ y) = ϕ(x) ∗ ϕ(y) for all x, y ∈ X.

Example 3.1. Consider X := {0, a, 1} with

Then (X; ∗, 0) is a d-algebra. If we define a map ϕ : X → X by ϕ(0) = 1, ϕ(a) = a and ϕ(1) = 0, then it is easy to see that ϕ is both self-inverse and order reversing, but it is not an anti-homomorphism, since ϕ(a∗1) = ϕ(0) = 1 and ϕ(1)∗ϕ(a) = 0∗a = 0.

Moreover, it is not a homomorphism, since ϕ(0 ∗ a) = ϕ(0) = 1 ≠ a = 1 ∗ a = ϕ(0) ∗ ϕ(a).

Proposition 3.2. Let (X, ∗, 0) be a d-algebra. If ϕ : X → X is a (anti-) homomorphism, then ϕ(0) = 0.

Proof. Since X is a d-algebra, by (D1), we obtain ϕ(0) = ϕ(x ∗ x) = ϕ(x) ∗ ϕ(x) = 0.

Proposition 3.3. If (X, ∗, 0) is a d-algebra, then every anti- homomorphism is order reversing.

Proof. Let ϕ : X → X be an anti-homomorphism. If we assume that x ∗ y = 0, then ϕ(y) ∗ ϕ(x) = ϕ(x ∗ y) = ϕ(0) = 0 by Proposition 3.2. This proves the proposition.

Remark. The converse of Proposition 3.3 need not be true in general. In Example 3.1, the mapping ϕ is an order reversing, but not an anti-homomorphism.

Let (X, ∗, 0) be a d-algebra and let ϕ : X → X be a map. We denote by 1 := ϕ(0).

Proposition 3.4. Let (X, ∗, 0) be a d-algebra and let ϕ : X → X be both order reversing and self-inverse. Then (X, ∗, 0) is bounded.

Proof. Given x ∈ X, we have

x ∗ 1 = x ∗ ϕ(0) [1 = ϕ(0)] = ϕ(ϕ(x)) ∗ ϕ(0) [ϕ: self-inverse] = 0 [ϕ: order reversing]

Let (X, ∗, 0) be a d-algebra. We define a relation “≤” on X by x ≤ y if and only if x ∗ y = 0 for all x, y ∈ X. Note that the relation ≤ need not be a partial order on X. We define a relation “∧ on X by x ∧ y := x ∗ (x ∗ y)) for all x, y ∈ X.

Proposition 3.5. Let (X, ∗, 0) be a d-algebra. If ϕ : X → X is self-inverse, then ϕ(1) = 0.

Proof. It follows from ϕ is self-inverse that 0 = ϕ(ϕ(0)) = ϕ(1).

Theorem 3.6. Let (X, ∗, 0) be a d-algebra and let ϕ : X → X be a self-inverse map. If we define x ∨ y := ϕ[ϕ(y) ∧ ϕ(x)], then

ϕ(x ∧ y) = ϕ(y) ∨ ϕ(x)

for all x, y ∈ X.

Proof. Given x, y ∈ X, we have

ϕ(x ∧ y) = ϕ[ϕ(ϕ(x)) ∧ ϕ(ϕ(y))] [ϕ: self-inverse] = ϕ[ϕ(a)) ∧ ϕ(b)] [a = ϕ(x), b = ϕ(y)] = b ∨ a = ϕ(y) ∨ ϕ(x)

Theorem 3.6 shows that the first De Morgan’s law implies the analog of the second De Morgan’s law and conversely, since x ∨ y ≠ y ∨ x in general. Moreover, it follows that x ∧ y = ϕ(ϕ(x ∧ y)) = ϕ[ϕ(y) ∨ ϕ(x)] for all x, y ∈ X.

Theorem 3.7. Let (X, ∗, 0) be a d-algebra with

for all x ∈ X. If ϕ : X → X is a self-inverse map, then x ∨ x = x and x ∧ x = x for all x ∈ X.

Proof. (i). Given x ∈ X, we have

x ∨ x = ϕ[ϕ(x) ∧ ϕ(x)] [Theorem 3.6] = ϕ[ϕ(x) ∗ (ϕ(x) ∗ ϕ(x)] = ϕ(ϕ(x) ∗ 0) [(D1)] = ϕ(ϕ(x)) [(1)] = x [ϕ: self-inverse]

(ii). x ∧ x = x ∗ (x ∗ x) = x ∗ 0 = x.

Proposition 3.8. Let (X, ∗, 0) be a d-algebra with

for all x, y, z ∈ X. Then x ∧ y ≤ x and x ∧ y ≤ y for all x, y ∈ X.

Proof. (i). Given x, y ∈ X, by applying (2), we obtain

(x ∧ y) ∗ a = (x ∗ (x ∗ y)) ∗ a = (x ∗ x) ∗ (x ∗ y) = 0 ∗ (x ∗ y) = 0

(ii). Given x, y ∈ X, we have (x∧y)∗y = (x∗(x∗y))∗y = (x∗y)∗(x∗y) = 0.

Theorem 3.9. Let (X, ∗, 0) be a d-algebra with the condition (2). If ϕ : X → X is a self-inverse anti-homomorphism, then x ∗ (x ∨ y) = 1 and y ∗ (x ∨ y) = 1 for all x, y ∈ X.

Proof. (i). Since ϕ : X → X is a self-inverse anti-homomorphism, for all x, y ∈ X, we obtain

x ∗ (x ∨ y) = x ∗ ϕ(ϕ(y) ∧ ϕ(x)) = x ∗ ϕ[ϕ(y) ∗ (ϕ(y) ∗ ϕ(x))] = ϕ(ϕ(x)) ∗ ϕ[ϕ(y) ∗ (ϕ(y) ∗ ϕ(x))] = ϕ[[ϕ(y) ∗ (ϕ(y) ∗ ϕ(x))] ∗ ϕ(x)] = ϕ[[(ϕ(y) ∗ ϕ(x)) ∗ (ϕ(y) ∗ ϕ(x))]] = ϕ(0) = 1

and

y ∗ (x ∨ y) = ϕ(ϕ(x)) ∗ ϕ[ϕ(y) ∗ (ϕ(y) ∗ ϕ(x))] = ϕ[[ϕ(y) ∗ (ϕ(y) ∗ ϕ(x))] ∗ ϕ(y)] = ϕ[(ϕ(y) ∗ ϕ(y)) ∗ (ϕ(y) ∗ ϕ(x))] = ϕ(0) = 1

 

CONCLUSION

Whether such functions exists or not depends on the special properties of the d-algebras. BCK-algebras have the partial order structure, but d-algebras have no such a structure and so we need to seek another conditions for obtaining the analog of structures in d-algebras. This kind of functional approach can be connected with mirror d-algebras discussed in [3] in a new direction.