Propiedades funcionales de las proteínas cárnicas

D(x1, x2) = 0 if and only if x1 =x2

and foremost, the triangle inequality, which is given by

D(x1, x2)≤D(x1, x3) +D(x2, x3). (10.1)

10.2

Main results

Inequality (10.1) is the main topic of this chapter. The other metric axioms are less difficult to verify. Since (10.1) describes the relation between three variables or objects instead of just two, some additional notation is required. Let

p111=P x11, 1 x2, 1 x3

denote the proportion of 1s shared by variablesx1,x2 andx3 in the same positions,

and let p110=P x11, 1 x2, 0 x3

denote the proportion of 1s shared by variables x1 and x2, and 0s by variable h3

in the same positions. With this notation we have that a =p11

12 = p111+p110. For

convenience, notation p111 _{will be used instead of P x}1 1, 1 x2, 1 x3 . The quantities a,

b, c, and d have subscripts

a12 =a(x1, x2)

b12 =b(x1, x2)

c12 =c(x1, x2)

d12 =d(x1, x2)

when comparing variables or objects x1 and x2. Furthermore, let D12 be short for

D(x1, x2). The subscripts are dropped whenever possible.

Theorem 10.1 covers the metric property for the relatively simple functions given by

DRR= 1−a and DSM=b+c.

Theorem 10.1. FunctionsDRR,DSMandD= 1−dsatisfy the triangle inequality

(10.1).

Proof: UsingDRR in (10.1) we obtain

1−a12≤1−a13+ 1−a23

2−2p111−p101−p011 ≥1−p111−p110

UsingD= 1−d and DSM in (10.1) we obtain respectively

1 +p001≥p000+p100+p010 (10.3) and

1 +p110+p001 ≥p111+p101+p011+p100+p010+p000. (10.4) (Interestingly, it does not suffice that for (10.4) to hold, both (10.2) and (10.3) are true). Inequalities (10.2), (10.3) and (10.4) are true because

1 =p111+p110+p101+p011+p100+p010+p001+p000. (10.5)

The proof of the metric property of DJac is less straightforward compared the

proof for coefficients considered in Theorem 10.1. The tool used is not adopted from Gower and Legendre (1986). Instead, the idea comes from Heiser and Bennani (1997), where it is used for three-way dissimilarities. The application below describes the tool for the simpler (two-way) case. In Chapter 18 a generalization of the proof of Theorem 10.2 is used. The next result shows that both

DJac= b+c a+b+c and D= b+c 1−a = b+c b+c+d

satisfy the triangle inequality.

Theorem 10.2. The functions DJac and

D= b+c

b+c+d

satisfy (10.1).

Proof: We consider the proof for DJac first. Adding p001 to both sides and p110 to

the left side of (10.5), we obtain

1 +p110+p001 ≥p111+p110+p101+p011+p100+p010+ 2p001+p000

which equals

(b13+c13) + (b23+c23)−(b12+c12)≥p001. (10.6)

DSM = 1−SSM and DJac are related by

DSM = (1−d12) b12+c12 1−d12 = (1−p000−p001)DJac. (10.7) Using (10.7) in (10.6) we obtain (1−p000) b13+c13 1−d13 +b23+c23 1−d23 −b12+c12 1−d12 ≥ p010 b13+c13 1−d13 +p100 b23+c23 1−d23 +p001 1− b12+c12 1−d12 .

10.2. Main results 113 Next, we consider the proof for D. Adding p110 _{to both sides and p}001 _{to the left}

side of (10.5), we obtain

(b13+c13) + (b23+c23)−(b12+c12)≥p110 (10.8)

instead of (10.6). DSM and D are related by

DSM= (1−a12) b12+c12 1−a12 = (1−p110−p111)D. (10.9) Using (10.9) in (10.8) we obtain (1−p111) b13+c13 1−a13 +b23+c23 1−a23 −b12+c12 1−a12 ≥ p101 b13+c13 1−a13 +p011 b23+c23 1−a23 +p110 1− b12+c12 1−a12 .

Since (1−p111_{)≥0 and D≤1, we conclude that D satisfies (10.1).}

This completes the proof.

Before studying any other coefficient, we note the following well-known result (see, for example, Gower and Legendre, 1986).

Theorem 10.3. Letebe a positive constant. IfDsatisfies(10.1),thenD/(e+D) satisfies (10.1). Proof: We have D12 e+D12 + D13 e+D13 ≥ D23 e+D23 if and only if e2(D12+D13−D23) + 2eD12D13+D12D13D23≥0.

Combining Theorem 10.3 with Theorem 10.1 or 10.2, various new results can be obtained. Consider the dissimilarities

DSS1 = 1−SSS1 = 2(b+c) a+ 2(b+c) = 2DJac 1 +DJac 2(b+c) 2(b+c) +d = 2D 1 +D where D= b+c b+c+d DRT = 1−SRT = 2(b+c) a+ 2(b+c) +d = 2DSM 1 +DSM .

Since DJac and DSM satisfy (10.1), application of Theorem 10.3 leads to the next

Proposition 10.1. The functions DSS3, DRT and

D= 2(b+c)

2(b+c) +d satisfy (10.1).

Next, it is shown what other members of

DGL1(θ) = 1−SGL1(θ) = 1−

a

(1−θ)a+θ(1−d), (10.10) apart from DJac and DSS1, satisfy the triangle inequality.

Theorem 10.4. The function DGL1(θ) satisfies (10.1) for 0< θ≤1.

Proof: By Theorem 10.2 DGL1(θ = 1) = DJac satisfies (10.1). For 0 < θ < 1, let

θ= (e+ 1)/e, wheree is a positive real number. Then (10.10) can be written as

DGL1(θ) = θDSM a+θDSM = (e+ 1)DSM ea+ (e+ 1)DSM . (10.11)

Dividing both numerator and denominator of (10.11) by 1−d we obtain

DGL1(θ) = (e+ 1)DJac eSJac+ (e+ 1)DJac = (e+ 1)DJac e+DJac . (10.12)

The right part of (10.12) satisfies (10.1) if and only ifDJac/(e+DJac) satisfies (10.1).

The result then follows from application of the Theorem 10.3.

10.3

Counterexamples

We finish the chapter with coefficients that do not satisfy the triangle inequality. For each coefficient, it suffices to present a counterexample (see also Gower and Legendre, 1986, Appendix II). Consider the three binary vectors

1 0 0 1 and 1 1 . We have DSS2 = 1− 2(a+d) 2a+b+c+ 2d →D12 = 1 andD13 =D23= 1 3 DGleas = 1− 2a p1+p2 →D12 = 1 andD13 =D23= 1 3 DDK= 1− a √ p1p2 →D12 = 1 andD13 =D23= 1− 1 √ 2 < 1 3 DKul = 1− a(p1+p2) 2p1p2 →D12 = 1 andD13 =D23= 1 4 DSim = 1− a min(p1, p2) →D12 = 1 andD13 =D23= 0.

10.4. Epilogue 115 Consider the three binary vectors

    1 0 0 0         0 1 0 0     and     1 1 0 0     . We have DCohen = 1− 2(ad−bc) p1q2+p2q1 →D12= 4 3 andD13=D23 = 1 2 DPhi = 1− ad−bc √ p1p2q1q2 →D12= 4 3 andD13=D23 = 1− 1 √ 3 < 1 2 DLoe = 1− ad−bc min(p1q2, p2q1) →D12= 4 3 andD13=D23 = 1 3. The dissimilarities do not satisfy the triangle inequality.

10.4

Epilogue

Only a few dissimilarities obtained with transformation D = 1−S turn out to be metric, that is, satisfy the triangle inequality. The key coefficients here are

DRR= 1−a =b+c+d and DSM= 1−a−d =b+c and DJac= 1− a a+b+c = b+c a+b+c.

Counterexamples were presented for various other coefficients. Since these two-way dissimilarities do not satisfy the triangle inequality, their multi-way formulations presented in Chapters 16 and 17 do not satisfy the generalizations of the triangle inequality considered in Part III of the thesis. Therefore, no metric properties of these coefficients are considered in Chapter 18.

Similarly to Chapters 7 and 8, it may be investigated if one of the functions that do not satisfy the triangle inequality in general, do satisfy the triangle inequality if the data matrix exhibits certain patterns or contains some form of structure. For example, if the data are Guttman vectors, the function

DDice = 1−

2a p1+p2

(10.13)

Proposition 10.2. Suppose thata12= min(p1, p2). Then DDice satisfies (10.1).

Proof: First, let p1 ≥p2 ≥p3. Using (10.13) in (10.1), we obtain

1 + 2p2 p1+p2 ≥ 2p3 p1 +p3 + 2p3 p2+p3 . (10.14) Equation (10.14) is true if (p1+p2)(p1+p3)(p2+p3)+2p2(p1+p3)(p2+p3) ≥ 2p3(p1+p2)(p2+p3)+2p3(p1+p2)(p1+p3) if and only if p2₁(p2−p3) + 3p1(p22 −p 2 3) +p2p3(p2−p3)≥0 (10.15)

holds. Sincep2 ≥p3, (10.15) is true.

Alternatively, let p3 ≥p2 ≥p1. Using (10.13) in (10.1), we obtain

1 + 2p1 p1+p2 ≥ 2p1 p1 +p3 + 2p2 p2+p3 . (10.16) Equation (10.16) is true if (p1+p2)(p1+p3)(p2+p2)+2p1(p1+p3)(p2+p3) ≥ 2p1(p1+p2)(p2+p3)+2p2(p1+p2)(p1+p3) if and only if p2₁(p3−p2) + 3p1(p23 −p 2 2) +p2p3(p3−p2)≥0 (10.17)

holds. Sincep3 ≥p2, (10.17) is true. This completes the proof.

Metric properties given a certain data structure may be investigated for other similarity coefficients as well. The applications of these coefficients would be very limited with respect to the general results for other coefficients in Section 10.2. Such results would be of theoretical interest only.

Part III

Multi-way metrics

CHAPTER

11

Axiom systems for two-way, three-way and

multi-way dissimilarities

Dissimilarities are functions that are used with various multivariate data analysis techniques. Well-known examples are multidimensional scaling and cluster analy- sis. A function is called a dissimilarity if it satisfies certain axioms, that is, it is nonnegative and symmetric, and it satisfies the axiom of minimality. In addition, a dissimilarity may satisfy axioms like the triangle inequality or the ultrametric in- equality. Dependencies between certain axioms have been noted by various authors (see, for example, Gower and Legendre (1986), Van Cutsem (1994) or Batagelj and Bren (1995) for the two-way case, and Joly and Le Calv´e (1995), Bennani-Dosse (1993) and Heiser and Bennani (1997) for the three-way case).

Although many authors (including the above-mentioned) point out that the used set of axioms do not form a system with a minimum number of axioms (due to de- pendencies between axioms), it remains (sometimes) unclear what this minimum set looks like. An axiom system can be a minimum set of axioms if it forms an independent system of axioms. Within an axiom system an axiom is called indepen- dent if it cannot be derived from the other axioms in the system. Another (perhaps more) important property of an axiom system is consistency. An axiom system is consistent if it lacks contradiction, that is, the ability to derive both a statement and its negation from a set of axioms.

In this chapter the axiom systems for two-way and three-way dissimilarities are studied. Some axioms for two-way dissimilarities were briefly considered in Section 1.2 and Section 10.1. To obtain axiom systems with a minimum number of axioms, the (known) dependencies between various axioms are reviewed. Next, consistency and independence of several axiom systems are established by means of simple mod- els. The remainder of the chapter is used to explore how basic axioms for multi-way dissimilarities, like nonnegativity, minimality and symmetry, may be defined. Gen- eralizations of the two-way metric and the three-way metrics are further studied in Chapter 12. Multi-way extensions of the three-way ultrametric inequalities are in- vestigated in Chapter 13. Using the tools for the axioms for three-way dissimilarities, independence and consistency may be established for the multi-way case.

11.1

Two-way dissimilarities

Let the functiond(x1, x2) : E×E → R assign a real number to each pair (x1, x2),

elements of the nonempty set E. The functiond(x1, x2) is called a two-way dissim-

ilarity between objectsx1 and x2 if it satisfies the axioms

(A1) d(x1, x2)≥0 (nonnegativity)

(A2) d(x1, x1) = 0 (minimality)

(A3) d(x1, x2) =d(x2, x1) (symmetry).

In the French literature, a dissimilarity d(x1, x2) is called respectively semi-proper

and proper if it satisfies

(A4) d(x1, x2) = 0 ⇒ d(x1, x3) =d(x2, x3) (evenness) (A5) d(x1, x2) = 0 ⇒ x1 =x2 (definiteness). Let p111₁₂₃=P x11, 1 x2, 1 x3

denote the proportion of 1s shared by variablesx1,x2 andx3 in the same positions,

let p110₁₂₃=P x11, 1 x2, 0 x3

denote the proportion of 1s shared by variables x1 and x2, and 0s by variable x3 in

the same positions, and let

p1₁ =Px11

denote the proportion of 1s in variable x1. For example, it holds that

11.1. Two-way dissimilarities 121 Proposition 11.1. (A1), (A2), (A3) and (A4)form a consistent and independent system of axioms. (A1), (A2), (A3) and (A5) form a consistent and independent system of axioms.

Proof: First, note that (A5) ⇒ (A4). Consistency of the two axiom systems is established by the first example of d(x1, x2) in the table below. The independence

of (A1), (A2) and (A3) with respect to the remaining four axioms is established with the bottom three examples ofd(x1, x2) in the table below.

Is the axiom valid?

d(x1, x2) (A1) (A2) (A3) (A4) (A5)

p1₁+p1₂−2p11₁₂ Yes Yes Yes Yes Yes 2p11₁₂−p1₁−p1₂ No Yes Yes Yes Yes

p1

1+p12−p1112 Yes No Yes Yes Yes

2p1₁+p1₂−3p11₁₂ Yes Yes No Yes Yes

Next, consider the functiond(x1, x2) = min(p11, p12)−p1112. It is readily verified that

d(x1, x2) satisfies (A1), (A2) and (A3). However, (A4) and (A5) are not valid if

there is a pair (x1, x2) for whichp1112 = min(p11, p12).

A two-way dissimilarityd(x1, x2) is called a distance if it satisfies definiteness and

(A6) d(x1, x2) ≤ d(x1, x3) +d(x2, x3) (triangle inequality).

A dissimilarity may also satisfy one of two axioms that define properties of trees, that is, an inequality by Buneman (1974)

(A7) d(x1, x2) +d(x3, x4) ≤ max[d(x1, x3) +d(x2, x4), d(x1, x4) +d(x2, x3)]

(additive tree) or

(A8) d(x1, x2) ≤ max[d(x1, x3), d(x2, x3)] (ultrametric inequality).

Proposition 11.2.

(i) (A6)together with (A2) ⇒(A1), (A3)and (A4) (ii) (A7)together with (A2) ⇒(A1), (A3), (A4) and (A6) (iii) (A8)together with (A2) ⇒(A1), (A3), (A4) and (A6).

Proof: The proof of (i) can be found in Gower and Legendre (1986, p. 6). For (ii) setting x3 equal to x4 in (A7) and applying (A2), we obtain (A6). For (iii), for

triplet (x1, x1, x2) we obtain d(x1, x2) ≥ 0, that is (A1). Moreover, (A8) together

Proposition 11.3. (A2), (A5)and (A6) (or (A7)or (A8)) form a consistent and independent system of axioms.

Proof: Consider the assertion with respect to (A6) first. An example for consistency is the function given by

d(x1, x2) = 1−p1112−p 00 12.

Validity of (A2) and (A5) is readily verified. Using d(x1, x2) in (A6) we obtain

1 +p11₁₂+p00₁₂≥p11₁₃+p00₁₃+p11₂₃+p00₂₃ if and only if 2p110₁₂₃+ 2p001₁₂₃ ≥0.

With respect to independence, consider the function d(x1, x2) = 1−p1112. Using

d(x1, x2) in (A6) we obtain

1 +p11₁₂≥p11₁₃+p11₂₃ if and only if p000₁₂₃+p100₁₂₃+p010₁₂₃+p001₁₂₃+ 2p110₁₂₃ ≥0.

Hence, d(x1, x2) satisfies (A6). Moreover, axiom (A5) is not violated. However, as

long as p1

1 6= 1, d(x1, x2) does not satisfy (A2). Hence, (A2) is independent from

(A5) and (A6).

Second, consider the functiond(x1, x2) = min(p11, p12)−p1112. Axiom (A2) is valid.

Assumingp1

1 ≥p12 ≥p13 and Usingd(x1, x2) in (A6), we obtain

2p1₃+p11₁₂ ≥p1₁+p11₁₃+p11₂₃ if and only if 2p001₁₂₃+p101₁₂₃ ≥p010₁₂₃.

Furthermore, (A5) is not valid if p11

12 = min(p11, p12) = p12 if and only if p0112 equals 0.

Thus, (A2) and (A6) may be valid, while (A5) is not.

Third, consider the functiond(x1, x2) = 2p1211−p11−p12. It is readily verified that

for this function (A2) and (A5) are valid. However, (A6) is only valid ifp110

123+p001123 ≤0

if and only ifp110

123 =p001123 = 0, since p110123 and p001123 are nonnegative quantities.

The proofs of the assertion with respect to (A7) and (A8) are very similar to that of (A6). Furthermore, suppose d(x1, x2) satisfies (A8). Then for the three two-way

dissimilarities defined on the same three objects, the largest two are equal. This property is unrelated to the value ofd(x1, x2).

11.2

Three-way dissimilarities

Axioms for three-way dissimilarities and distances can be found in Bennani-Dosse (1993), Heiser and Bennani (1997) and Chepoi and Fichet (2007). In addition, three-way distances are considered in Joly and Le Calv´e (1995). Let d3(x1, x2, x3) :

E×E×E →_Rbe a function that assigns a real number to each triplet (x1, x2, x3).

Heiser and Bennani (1997, p. 191) call d3(x1, x2, x3) a three-way dissimilarity if it

satisfies the axioms

(B1a) d3(x1, x2, x3)≥0 (nonnegativity)

(B2a) d3(x1, x1, x1) = 0 (minimality)

(B3) d3(x1, x2, x3) = d3(x1, x3, x2) =d3(x2, x1, x3) =

11.2. Three-way dissimilarities 123 the three-way generalizations of (A1), (A2) and (A3), and in addition

d3(x1, x1, x2) = d3(x1, x2, x2). (11.1)

Equality (11.1) is referred to as the diagonal-plane equality by Heiser and Bennani (1997), and is also proposed in Joly and Le Calv´e (1995).

Equality (11.1) is an answer to a complication that arises with three-way dissim- ilarities, not encountered with two-way dissimilarities, when one of three variables or entities is identical to one of the others. For this reason, Chepoi and Fichet (2007) studied explicitly the case of three-way dissimilarities for which all entities are different. The lack of resemblance between the two nonidentical entities should, according to Heiser and Bennani (1997), remain invariant regardless of which two entities are the same:

d3(x1, x1, x2) = d3(x1, x2, x2) =d3(x1, x2, x1) =

d3(x2, x1, x1) = d3(x2, x1, x2) =d3(x2, x2, x1).

Equality (11.1) is referred to as the diagonal-plane equality in Heiser and Bennani (1997), because it requires equality of the three matrices

{d3(x1, x1, x2)}, {d3(x1, x2, x2)} and {d3(x1, x2, x1)}

which are formed by cutting the three-way cube or block diagonally, starting at one of the three edges joining at the node or corner d(1,1,1). This seems to be a misnomer, since equality (11.1) only requires equality of the first two matrices. Equality (11.1) together with three-way symmetry (B3) implies the stronger equality

(B4) d3(x1, x1, x2) = d3(x1, x2, x2) =d3(x1, x2, x1).

Proposition 11.4. (B1a), (B2a), (B3)and (B4)form a consistent and indepen- dent system of axioms.

Proof: Consistency of the axiom system is shown with the first example of

d3(x1, x2, x3) in the table below.

Is the axiom valid?

d3(x1, x2, x3) (B1a) (B2a) (B3) (B4)

1−p111

123−p000123 Yes Yes Yes Yes

p111

123+p000123−1 No Yes Yes Yes

1−p111₁₂₃ Yes No Yes Yes

p1

1−p111123 Yes Yes No Yes

p1

1+p12 +p13−3p111123 Yes Yes Yes No

Independence is established with the bottom four examples of d3(x1, x2, x3) in the

At this point it should be noted that there exists mathematical literature on multi-way concepts, including distances and metrics, that is older that the above mentioned literature. Some of the references from this literature may be found in Deza and Rosenberg (2000, 2005). Characteristic of this literature are the extensions of axioms (A1) and (A2) given by

(B1b) x1 6=x2 ⇒d3(x1, x2, x3)>0 for somex3 ∈E

(B2b) d3(x1, x1, x2) = 0

and axiom (B6c) presented below. Axiom (B2b) makes perfect sense in geometry whered3(x1, x1, x2) is, for example, the area of the triangle with verticesx1,x2, and

x3. Deza and Rosenberg (2000, 2005) find axioms (B1b) and (B2b) too restrictive

and drop them. The two axioms are also ignored in this chapter.

A three-way dissimilarity d3(x1, x2, x3) is called a three-way distance in Heiser

and Bennani (1997, p. 191) if it satisfies

(B5) d3(x1, x2, x3) = 0 ⇒ x1 =x2 =x3 (definiteness)

and the so-called tetrahedral inequality

(B6a) 2d3(x1, x2, x3)≤d3(x2, x3, x4) +d3(x1, x3, x4) +d3(x1, x2, x4).

Alternatively, Joly and Le Calv´e (1995) call d(x1, x2, x3) a three-way distance if it

satisfies

(B6b) d3(x1, x2, x3)≤d3(x2, x3, x4) +d3(x1, x3, x4)

(B7) d3(x1, x2, x3)≥d3(x1, x1, x3)

and a proper three-way distance if it, in addition, satisfies (B5). Axioms (B6a) and (B6b) are called respectively strong and weak metrics in Chepoi and Fichet (2007). Deza and Rosenberg (2000, 2005) present yet another extension of the triangle inequality. The so-called tetrahedron inequality is given by

(B6c) d3(x1, x2, x3)≤d3(x2, x3, x4) +d3(x1, x3, x4) +d3(x1, x2, x4).

Axiom (B6c) is not studied further in this chapter (but see Chapter 12).

Three-way generalizations of two-way ultrametric inequality (A8) are considered in Joly and Le Calv´e (1995, p. 195) and Bennani-Dosse (1993, p. 99-110):

(B8a) d3(x1, x2, x3)≤max [d3(x2, x3, x4), d3(x1, x3, x4)]

(B8b) d3(x1, x2, x3)≤max [d3(x2, x3, x4), d3(x1, x3, x4), d3(x1, x2, x4)].

Axioms (B8a) and (B8a) are called respectively strong and weak ultrametrics in Chepoi and Fichet (2007).

11.2. Three-way dissimilarities 125 As noted in Bennani-Dosse (1993, p. 20), the dependencies between (B1) to (B8) are not as straightforward as the dependencies between (A1) to (A8) given in Proposition 11.2.

Proposition 11.5.

(B6b) together with (B7) and (B2a) ⇒ (B1a) (i) (B6b) together with (B3)⇒ (B1a)

(B6a) together with (B3) ⇒ (B1a) and (B6b) (B7) together with (B3) ⇒(B4)

(ii) (B8a) ⇒ (B6a), (B7) and (B8b).

The proofs for (i) and (ii) are presented below. The proofs of the other assertions can be found in Joly and Le Calv´e (1995, p. 193) and Heiser and Bennani (1997, p. 192).

Proof: For (i), adding the two variants of (B6b)

d3(x1, x2, x3)≤d3(x2, x3, x4) +d3(x1, x3, x4)

and d3(x2, x3, x4)≤d3(x1, x2, x3) +d3(x1, x3, x4)

we obtain 2d3(x1, x3, x4)≥0. With respect to (ii), note that, ifd(x1, x2, x3) satisfies

(B8a), then for any four three-way dissimilarities the largest three are equal. The dependencies in Proposition 11.5 suggest the independence of various axiom systems. First, we consider a system of structural, that is, non-metric axioms. Proposition 11.6. (B1a), (B2a), (B3), (B5) and (B7) form a consistent and independent system of axioms.

Proof: An example of consistency of the axiom system is the function

d3(x1, x2, x3) = 1−p111123−p000123. It is readily verified that (B1a), (B2a), (B3) and

(B5) are valid. Using d3(x1, x2, x3) in (B7) we obtain

p11₁₃+p00₁₃ ≥p111₁₂₃+p000₁₂₃ if and only if p101₁₂₃+p010₁₂₃ ≥0.

With respect to independence, consider the functiond3(x1, x2, x3) = 3p111123−p11−p12−

p1

3. Axioms (B2a), (B3) and (B5) are valid, but (B1a) is not. Using the function

in (B7) we obtain

3p111₁₂₃+p1₁ ≥3p11₁₃+p1₃

p100₁₂₃+p110₁₂₃ ≥3p101₁₂₃+p001₁₂₃+p011₁₂₃ p10₁₃≥3p101₁₂₃+p01₁₃.

Second, consider the functiond3(x1, x2, x3) =p11+p12+p13−2p111123. Axioms (B1a),

(B3) and (B5) are valid, but (B2a) is not. The function satisfies (B7) if and only ifp01₁₂+ 2p101₁₂₃ ≥p10₁₂. Thus, axiom (B2a) is independent from (B1a), (B3), (B5) and (B7).

Third, consider the functiond3(x1, x2, x3) = 2p11+p12+p13−4p111123. Axioms (B1a),

(B2a) and (B5) are valid, but (B3) is not. The function satisfies (B7) if and only if p01

12+ 4p101123 ≥p1012, which shows that (B3) is independent from the remaining four

axioms.

Next, consider the function

d3(x1, x2, x3) = min(p1112, p 11 13, p 11 23)−p 111 123.

It is readily verified that (B1a), (B2a), (B3) and (B7) are valid. However, if there is a triple (x1, x2, x3) for which p111123 = min(p1112, p1113, p1123), then (B5) does not hold.

Finally, consider the function d3(x1, x2, x3) = p11 +p12 +p13 −3p111123. It is read-

ily verified that (B1a), (B2a), (B3) and (B5) are valid. Furthermore, we have

d3(x1, x2, x3)≤ d3(x1, x1, x2) if and only if p0112+ 3p101123 ≤ p1012, which show the inde-

pendence of (B7) with respect to the remaining four axioms.

Finally, we consider an axiom system with a minimum number of axioms.

Proposition 11.7. (B2a), (B3), (B5), (B6a) and (B7) form a consistent and independent system of axioms.

Proof: An example for the consistency of the axiom system is the function

d3(x1, x2, x3) = 1−p123111 −p000123. It is readily verified that (B2a), (B3), (B5) and

(B7) are valid. Using d3(x1, x2, x3) in (B6a) we obtain

1−(p111₂₃₄+p₁₃₄111+p111₁₂₄+p000₂₃₄+p000₁₃₄+p000₁₂₄) + 2p₁₂₃111+ 2p000₁₂₃≥0. (11.2) Since the quantity in between brackets in (11.2) is smaller than unity, (B6a) is valid. With respect to independence, consider the function d3(x1, x2, x3) = p11 +p12 +

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