AN EXAMPLE OF A COMMUTATIVE BASIC

DOI: 10.2478/s12175-010-0003-0 Math. Slovaca 60 (2010), No. 2, 171–178

AN EXAMPLE OF A COMMUTATIVE BASIC ALGEBRA WHICH IS NOT AN MV-ALGEBRA Michal Botur (Communicated by Anatolij Dvureˇ censkij ) ABSTRACT. Many algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algeˇ R.: Finite commutative basic bras. In the recent papers [BOTUR, M.—HALAS, algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. ˇ R.: Complete commutative basic algebras, Order 24 and [BOTUR, M.—HALAS, (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOˇ R.: Complete commutative basic algebras, Order 24 (2007), TUR, M.—HALAS, 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra. c
2010 Mathematical Institute Slovak Academy of Sciences

1. Introduction MV-algebras were introduced by C. C. Chang [Cha] as an algebraic counterpart of the many-valued propositional L
ukasiewicz logic. Recall that an MV-algebra is an algebra A = (A, ⊕, ¬, 0) of type (2, 1, 0) where (A, ⊕, 0) is a commutative monoid satisfying the identities (1) ¬¬x = x (2) x ⊕ ¬0 = ¬0 (3) ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x. 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 06D35, 06F35; Secondary 03G10. K e y w o r d s: basic algebra, lattice with sectional antitone involutions, MV-algebra. This work was supported by the Research and Development Council of the Czech Government via the project MSM6198959214.

Unauthenticated Download Date | 9/25/15 2:01 AM

MICHAL BOTUR

As known, if A = (A, ⊕, ¬, 0) is an MV-algebra then
(A) = (A, ∨, ∧, 0, 1), where x → y := ¬x ⊕ y x ∨ y := ¬(¬x ⊕ y) ⊕ y, x ∧ y := ¬(¬x ∨ ¬y), 1 := ¬0, is a bounded distributive lattice whose induced order, the so-called natural order of an MV-algebra A, is given by x ≤ y ⇐⇒ ¬x ⊕ y = 1. In addition, given an interval (called section) [a, 1] of an MV-algebra A and x ∈ [a, 1], the mapping a : [a, 1] −→ [a, 1] given by xa := ¬x ⊕ a is known to be an antitone involution on the interval [a, 1]. This fact lead us to the following concepts, see [CHK1]: A lattice with sectional antitone involutions is a system L = (L; ∨, ∧, (a )a∈L , 0, 1), where (L; ∨, ∧, 0, 1) is a bounded lattice such that every principal orderfilter [a, 1] (which is called a section) possesses an antitone involution x → xa . By a basic algebra we shall mean an algebra A = (A; ⊕, ¬, 0) of type (2, 1, 0) satisfying the identities (1)–(3) and (BA) ¬(¬(¬(x ⊕ y) ⊕ y) ⊕ z) ⊕ (x ⊕ z) = 1. Remarkably, basic algebras and lattices with sectional antitone involutions are in a one-to-one correspondence. Namely, given a lattice with section antitone involutions (L, ∨, ∧, (a )a∈L , 0, 1), the algebra (A, ⊕, ¬, 0), where x⊕y := (x0 ∨y)y and ¬x := x0 is a basic algebra. Conversely, for a basic algebra (A, ⊕, ¬, 0), the structure (A, ∨, ∧, (a )a∈L , 0, 1) with induced order given by x ≤ y ⇐⇒ ¬x ⊕ y = 1 a and with x := ¬x ⊕ a for a ≤ x is a lattice with sectional antitone involutions. Repeated application of the two constructions results in the original structure. Given a basic algebra A and x, y ∈ A, the elements x, y are said to commute if x⊕y = y⊕x. If every two elements of A commute then A is called a commutative basic algebra. It is easy to check (see [CHK1]) that the commutativity of ⊕ is equivalent to the contraposition law x → y = ¬y → ¬x. Commutative basic algebras play an important role in the structural theory of basic algebras. More precisely, every basic algebra A is a union of so-called blocks, maximal sets of pairwise commuting elements. It has been proved in [BoHa1] that every finite commutative basic algebra is an MV-algebra; in other words, in the finite case, the commutativity of ⊕ yields its associativity. In [BoHa2] complete commutative basic algebras (i.e. those for which the induced lattice is complete) were studied. We have shown that every complete commutative basic algebra is a subdirect product of chain basic algebras. Moreover, as discussed in [BoHa2], in the complete case it is enough to study commutative basic algebras on the interval [0, 1] of reals. 172 Unauthenticated Download Date | 9/25/15 2:01 AM

A COMMUTATIVE BASIC ALGEBRA WHICH IS NOT AN MV-ALGEBRA

In conclusion, in [BoHa2] we posed the following open problem: (P) Is every commutative basic algebra on the interval [0, 1] of reals an MV-algebra? The aim of this paper is to present a negative solution of the above problem. Namely, we present a commutative basic algebra on the interval [0, 12] of the reals which is not an MV-algebra. Thus, the variety of MV-algebras is a proper subvariety of the variety of commutative basic algebras.

2. Commutative basic algebras on an interval of the reals Throughout this section we will work on the interval [0, 12] of reals. Firstly, recall that x → 0 = ¬x and (x → 0) → y = ¬x → y = x ⊕ y. Thus the operations ⊕ and ¬ are uniquely determined by → and 0. Moreover, for x ≤ y we have x → y = 12 (12 is the greatest element).


 1

(a) Let A = ([0, 12], ⊕, ¬, 0) be a commutative basic algebra. Then the function f : [0, 12]2 −→ [0, 12], where f (x, y) := x → y is continuous (in the usual sense) and for all y ∈ [0, 12] and all x, x1 , x2 ∈ [y, 12] we have (i) f (x, y) = f (¬y, ¬x) (ii) if x ≥ y then f (f (x, y), y) = x (iii) if x1 ≤ x2 then f (x1 , y) ≥ f (x2 , y). (b) Conversely, if f : [0, 12]2 −→ [0, 12] is a continuous function satisfying (i)–(iii) of (a), then A = ([0, 12], ⊕, ¬, 0), where ¬x := f (x, 0) and x ⊕ y := f (¬x, y) is a commutative basic algebra. P r o o f. (a)(i) f (x, y) = x → y = ¬y → ¬x = f (¬y, ¬x) by the contraposition law. (ii) If x ≥ y then f (f (x, y), y) = (xy )y = x. (iii) If x1 ≤ x2 then f (x1 , y) = (x1 )y ≥ (x2 )y = f (x2 , y). Since fy (x) := f (x, y) is an antitone involution on [y, 1], thus a monotonously decreasing bijection on [y, 1] and fy (x) = 1 if x ≤ y, it is continuous with respect to x. Moreover, due to equalities f (x, y) = f (¬y, ¬x) = f (f (y, 0), f (x, 0)), f is continuous with respect to y. (b) By (ii) and (iii), the function fy (x) := f (x, y) is antitone involution on the interval [y, 1], hence A is a basic algebra. That A is commutative easily follows by (i) which is exactly the contraposition law (see [BoHa1]).
We can see that for any x ∈ [0, 12] there exists a unique x∗ ∈ [0, 12] such that x∗ → x = x∗ . In other words, x∗ is the fixpoint of the involution x .


 2 Any basic algebra A = ([0, 12], ⊕, ¬, 0) is isomorphic to a basic algebra A = ([0, 12], ⊕ , ¬ , 0) such that ¬ x = 12 − x for any x ∈ [0, 12]. 173 Unauthenticated Download Date | 9/25/15 2:01 AM

MICHAL BOTUR

P r o o f. Let α∗ : [0∗ , 12] −→ [6, 12] be any order isomorphism. Define the mapping
if x ∈ [0∗ , 12] α∗ (x) α(x) := 12 − α∗ (¬x) otherwise. Clearly the mapping α is a bijection on the interval [0, 12]. If x ∈ [0, 0∗ ] then ¬x ∈ [0∗ , 12], so we have α(x) = 12 − α∗ (¬x) and also α(¬x) = α∗ (¬x). Clearly this yields ¬ α(x) = α(¬x). Analogously in the second case. Further denote for any x, y ∈ A x → y = α(α−1 (x) → α−1 (y)). One can easily prove that defining x ⊕ y := ¬x → y, A = ([0, 12], ⊕, ¬ , 0) is a commutative basic algebra for which A ∼ = A , the isomorphism of which is given by α.
Thus without loss of generality, we may suppose that the operation ¬ is defined by ¬x := 12 − x. Now, consider the following sets:   g = x∗ , x : x ∈ [0, 12] ,   h = 12 − x, 12 − x∗ : x ∈ [0, 12] ,   k = x, ¬x : x ∈ [6, 12] . The continuity of f yields that g, h, k are continuous curves. Moreover, they divide the area { x, y ∈ [0, 12]2 : x ≥ y} into six parts, as visualized in Fig. 2.1. Remark that f , g, h need not be straight lines in general, but in our example this is the case. As well-known, the implication →M V on [0, 12] considered as an MV-algebra is given by stipulation x →M V y := 12−x+y. Consider another function f (x, y) of the form: f (x, y) := 12 − x + y + d(x, y), where d(x, y) measures the “difference” of f (x, y) and “x →M V y”. The idea of constructing a commutative basic algebra which is not an MV-algebra is based on finding the non-zero function d(x, y). Hence, given f (x, y) as before, we derive the properties of d(x, y).


 3 For all x, y ∈ [0, 12] with x ≥ y we have d(x, y) = d(¬y, ¬x) = d(f (x, y), y). P r o o f. We have 12 − x + y + d(x, y) = f (x, y) = f (¬y, ¬x) = 12 − (12 − y) + (12 − x) + d(¬y, ¬x) = 12 − x + y + d(¬y, ¬x). From this d(x, y) = d(¬y, ¬x) follows immediately. Moreover, we know that x = f (f (x, y), y) for x ≥ y. Thus x = 12−f (x, y)+y +d(f (x, y), y) = 12−(12−x+y +d(x, y))+y +d(f (x, y), y) = x − d(x, y) + d(f (x, y), y), from which we obtain d(x, y) = d(f (x, y), y).
Further, consider the areas I.-VI. according to Figure 2.1, defined by the curves g, h, k. The Table 2.1 describes the membership of points into the above areas. 174 Unauthenticated Download Date | 9/25/15 2:01 AM

A COMMUTATIVE BASIC ALGEBRA WHICH IS NOT AN MV-ALGEBRA

Figure 2.1 Table 2.1

x, y

¬y, ¬x

f (x, y), y

I. II. III. IV. V. VI.

II. I. VI. V. IV. III.

IV. III. II. I. VI. V.

From this we can see that if d(x, y) is defined on one of the areas I.–VI., then it is defined on the remaining ones uniquely.

3. Example of a commutative basic algebra which is not an MV-algebra Now we present a commutative basic algebra A = ([0, 12], ⊕, ¬, 0) by constructing the function f (x, y). Let the curves g, h, k be defined as follows: • g is the straight line from 6, 0 to 12, 12 175 Unauthenticated Download Date | 9/25/15 2:01 AM

MICHAL BOTUR Table 3.1

I. II. III. IV. V. VI.

M

N

O


0; 0,
6; 6,
6; 4
6; 6,
8; 6,
12; 12
12; 6,
12; 12,
11.8; 6
12; 0,
11.8; 4,
12; 6
6; 0,
12; 0,
8; 0.2
0; 0,
6; 0,
6; 0.2


0; 0,
6; 4,
8; 4
8; 4,
8; 6,
12; 12
8; 4,
12; 12,
11.8; 6
12; 0,
8; 4,
11.8; 4
12; 0,
8; 4,
8; 0.2
0; 0,
8; 4,
6; 0.2


6; 4,
8; 4,
6; 6
8; 4,
8; 6,
6; 6
8; 4,
12; 6,
11.8; 6
8; 4,
12; 6,
11.8; 4
6; 0,
8; 4,
8; 0.2
6; 0,
8; 4,
6; 0.2

Table 3.2

f (x, y) I. II. III. IV. V. VI.

M

N

O

12 − 0.1x + 0.1y 12 − 0.1x + 0.1y 120 − 10x + y 120 − 10x + y 12 − x + 10y 12 − x + 10y

12 − 1.9x + 2.8y 22.8 − 2.8x + 1.9y 12·1.9 1 1.9 2.8 − 2.8 x + 2.8 y 12 1 2.8 1.9 − 1.9 x + 1.9 y 12·2.8 2.8 1 1.9 − 1.9 x + 1.9 y 1 12 − 1.9 2.8 x + 2.8 y

22.8 − 1.9x + 0.1y 1.2 − 0.1x + 1.9y 12 − 10x + 19y 1 1 12 − 1.9 x + 19 y 12 1 1 − x + 1.9 19 1.9 y 120 − 19x + 10y

• h is the straight line from 0, 0 to 6, 12 • k is the straight line from 6, 6 to 12, 0 . Thus, the curves g, h, k coincide on the standard MV-algebra on [0, 12]. We can see that the function d(x, y) is zero on these curves as well as on the border of { x, y ∈ [0, 12]2 : x ≥ y}. In the area I. (see Figure 2.1) we define the function d(x, y) as a “pyramid” with vertex in 6, 4 and with height 1.8. So, the area I. is splitting into three subareas M , N , O (see Figure 3.1 and Table 3.1, first row) and the function d(x, y) we will defined piecewise. Thus, on the area I. we have:  0.9x − 0.9y if x, y ∈ M 1.8y − 0.9x if x, y ∈ N d(x, y) := 10.8 − 0.9x − 0.9y if x, y ∈ O. According to Lemma 3, d(x, y) is then defined on areas II.-VI uniquely. Thus, each area I.–VI. is splitting into the three subareas M , N , O. Clearly, any of these areas is a triangle. Finally we obtain the function f (x, y) defined piecewise on each of the eighteen areas (in Table 3.1 we have defined areas and in Table 3.2 we have the relevant terms).

176 Unauthenticated Download Date | 9/25/15 2:01 AM

A COMMUTATIVE BASIC ALGEBRA WHICH IS NOT AN MV-ALGEBRA

Figure 3.1

Clearly, if x ≤ y then f (x, y) = 12. The graph of the function f (x, y) is visualized on Figure 3.2. One can easily check that f (x, y) is continuous and f (x, y) has the following values on the boundaries of the areas: • 8 → 4 = 11.8 → 6 = 6 → 0.2 = 8 • 6 → 4 = 8 → 6 = 11.8 • 11.8 → 4 = 8 → 0.2 = 12 → 6 = 6 → 0 = 6 • 12 → 12 = 6 → 6 = 0 → 0 = 12 • 12 → 0 = 0. Since f (x, y) is piecewise linear, so these are also f (f (x, y), y) and f (¬y, ¬x). It can be checked that f (f (x, y), y) = x and f (¬y, ¬x) = f (x, y) on all of the boundaries of areas I.–VI., hence f (x, y) fulfils these identities everywhere. Thus, if we denote x ⊕ y = ¬x → y = f (¬x, y) ¬x = f (x, 0) = 12 − x then A = ([0, 12], ⊕, ¬, 0) is a commutative basic algebra. Finally, one can . compute that 10 → (8 → 4) = 10 → 8 = 10 whereas 8 → (10 → 4) = 8 → . 6.95 = 11.89. Thus in A the exchange identity does not hold and A is not an MV-algebra. 177 Unauthenticated Download Date | 9/25/15 2:01 AM

MICHAL BOTUR

Figure 3.2

REFERENCES ˇ R.: Complete commutative basic algebras, Order 24 (2007), [BoHa2] BOTUR, M.—HALAS, 89–105. ˇ R.: Finite commutative basic algebras are MV-effect algebras, [BoHa1] BOTUR, M.—HALAS, J. Mult.-Valued Logic Soft Comput. 14 (2008), 69–80. [Ch1] CHAJDA, I.: Lattices and semilattices having an antitone involution in every upper interval, Comment. Math. Univ. Carolin. 44 (2003), 577–585. ´ P.: Bounded lattices with antitone involutions and [CE] CHAJDA, I.—EMANOVSKY, properties of MV-algebras, Discuss. Math. Gen. Algebra Appl. 24 (2004), 31–42. ˇ R.—KUHR, ¨ [CHK1] CHAJDA, I.—HALAS, J.: Distributive lattices with sectionally antitone involutions, Acta Sci. Math. (Szeged) 71 (2005), 19–33. ˇ R.—KUHR, ¨ [CHK2] CHAJDA, I.—HALAS, J.: Many-valued quantum algebras, Algebra Universalis 60 (2009), 63–90. [Cha] CHANG, C. C.: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 464–490. [CDM] CIGNOLI, R. L. O.—D’OTTAVIANO, M. L.—MUNDICI, D.: Algebraic Foundations of Many-valued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. ˇ [DvPu] DVURCENSKIJ, A.—PULMANNOV´ a, S.: New Trends in Quantum Scructures, Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava 2000. [Gi] GIUNTINI, R.: Quantum MV-algebras, Studia Logica 56 (1996), 303–417. Received 16. 1. 2008 Accepted 5. 8. 2008

Department of Algebra and Geometry Palack´ y University Olomouc Tomkova 40 CZ–779 00 Olomouc Czech REPUBLIC E-mail : [email protected]

178 Unauthenticated Download Date | 9/25/15 2:01 AM