RATIO MATHEMATICA 22 (2012) 3-12 ISSN:1592-7415

About a definition of metric over an
abelian linearly ordered group

Bice Cavallo, Livia D’Apuzzo
University Federico II, Naples, Italy

bice.cavallo@unina.it, liviadap@unina.it

Abstract

A G-metric over an abelian linearly ordered group G = (G,⊙,≤)
is a binary operation, dG, verifying suitable properties. We consider
a particular G metric derived by the group operation ⊙ and the total
weak order ≤, and show that it provides a base for the order topology
associated to G.

Key words: G-metric, abelian linearly ordered group, multi-criteria
decision making.

2000 AMS subject classifications: 06F20, 06A05, 90B50.

1 Introduction

The object of the investigation in our previous papers have been the pair-
wise comparison matrices that, in a Multicriteria Decision Making context,
are a helpful tool to determine a weighted ranking on a set X of alternatives
or criteria [1], [2], [3]. The pairwise comparison matrices play a basic role in
the Analytic Hierarchy Process (AHP), a procedure developed by T.L. Saaty
[17], [18], [19]. In [14], the authors propose an application of the AHP for
reaching consensus in Multiagent Decision Making problems; other consensus
models are proposed in [6], [11], [15], [16].

The entry aij of a pairwise comparison matrix A = (aij) can assume
different meanings: aij can be a preference ratio (multiplicative case) or a
preference difference (additive case) or aij is a preference degree in [0, 1]
(fuzzy case). In order to unify the different approaches and remove some
drawbacks linked to the measure scale and a lack of an algebraic structure,

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B. Cavallo and L. D’Apuzzo

in [7] we consider pairwise comparison matrices over abelian linearly ordered
groups (alo-groups). Furthermore, we introduce a more general notion of
metric over an alo-group G = (G,⊙,≤), that we call G-metric; it is a binary
operation on G

d : (a, b) ∈ G2 → d(a, b) ∈ G,

verifying suitable conditions, in particular: a = b if and only if the value
of d(a, b) coincides with the identity of G. In [7], [8], [9], [10] we consider a
particular G-metric, based upon the group operation ⊙ and the total order
≤. This metric allows us to provide, for pairwise comparison matrices over
a divisible alo-group, a consistency index that has a natural meaning and it
is easy to compute in the additive and multiplicative cases.

In this paper, we focus on a particular G-metric introduced in [7] look-
ing for a topology over the alo-group in which the G-metric is defined. By
introducing the notion of dG-neighborhood of an element in an alo-group
G = (G,⊙,≤), we show that the above G-metric generates the order topol-
ogy that is naturally induced in G by the total weak order ≤.

2 Abelian linearly ordered groups

Let G be a non empty set, ⊙ : G × G → G a binary operation on G, ≤
a total weak order on G. Then G = (G,⊙,≤) is an alo-group, if and only if
(G,⊙) is an abelian group and

a ≤ b ⇒ a⊙ c ≤ b⊙ c. (1)

As an abelian group satisfies the cancellative law, that is a⊙c = b⊙c ⇔
a = b, (1) is equivalent to the strict monotonicity of ⊙ in each variable:

a < b ⇔ a⊙ c < b⊙ c. (2)

Let G = (G,⊙,≤) be an alo-group. Then, we will denote with:

• e the identity of G;

• x(−1) the symmetric of x ∈ G with respect to ⊙;

• ÷ the inverse operation of ⊙ defined by a÷ b = a⊙ b(−1),

• x(n), with n ∈ N0, the (n)-power of x ∈ G:

x(n) =

{
e, if n = 0
x(n−1) ⊙x, if n ≥ 1;

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About a definition of metric over an abelian linearly ordered group

• < the strict simple order defined by x < y ⇔ x ≤ y and x ̸= y;

• ≥ and > the opposite relations of ≤ and < respectively.

Then

b(−1) = e÷ b, (a⊙ b)(−1) = a(−1) ⊙ b(−1), (a÷ b)(−1) = b÷a; (3)

moreover, assuming that G is no trivial, that is G ̸= {e}, by (2) we get

a < e ⇔ a(−1) > e, a < b ⇔ a(−1) > b(−1) (4)

a⊙a > a ∀a > e, a⊙a < a ∀a < e. (5)

By definition, an alo-group G is a lattice ordered group [4], that is there
exists a ∨ b = max{a, b}, for each pair (a, b) ∈ G2. Nevertheless, by (5), we
get the following proposition.

Proposition 2.1. A no trivial alo-group G = (G,⊙,≤) has neither the great-
est element nor the least element.

Order topology. If G = (G,⊙,≤) is an alo-group, then G is naturally
equipped with the order topology induced by ≤ that we will denote with τG.
An open set in τG is union of the following open intervals:

• ]a, b[= {x ∈ G : a < x < b};

• ] ←, a[= {x ∈ G : x < a};

• ]b,→ [= {x ∈ G : x > b};

and a neighborhood of c ∈ G is an open set to which c belongs. Then G×G is
equipped with the related product topology. We say that G is a continuous
alo-group if and only if ⊙ is continuous.

Isomorphisms between alo- groups An isomorphism between two alo-
groups G = (G,⊙,≤) and G′ = (G′,◦,≤) is a bijection h : G → G′ that is
both a lattice isomorphism and a group isomorphism, that is:

x < y ⇔ h(x) < h(y) and h(x⊙y) = h(x)◦h(y). (6)

Thus, h(e) = e′, where e′ is the identity in G′, and

h(x(−1)) = (h(x))(−1). (7)

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B. Cavallo and L. D’Apuzzo

By applying the inverse isomorphism h−1 : G′ → G, we get:

h−1(x′ ◦y′) = h−1(x′)⊙h−1(y′), h−1(x′
(−1)

) = (h−1(x′))(−1). (8)

By the associativity of the operations ⊙ and ◦, the equality in (6) can be
extended by induction to the n-operation

⊙n
i=1 xi, so that

h(
n⊙

i=1

xi) = ⃝ni=1h(xi), h(x
(n)) = h(x)(n). (9)

3 G-metric
Following [5], we give the following definition of norm:

Definition 3.1. Let G = (G,⊙,≤) be an alo-group. Then, the function:

|| · || : a ∈ G →||a|| = a∨a(−1) ∈ G (10)

is a G-norm, or a norm on G.

The norm ||a|| of a ∈ G is also called absolute value of a in [4].

Proposition 3.1. [7] The G-norm satisfies the properties:

1. ||a|| = ||a(−1)||;

2. a ≤ ||a||;

3. ||a|| ≥ e;

4. ||a|| = e ⇔ a = e;

5. ||a(n)|| = ||a||(n);

6. ||a⊙ b|| ≤ ||a||⊙ ||b||. (triangle inequality)

Definition 3.2. Let G = (G,⊙,≤) be an alo-group. Then, the operation

d : (a, b) ∈ G2 → d(a, b) ∈ G

is a G-metric or G-distance if and only if:

1. d(a, b) ≥ e;

2. d(a, b) = e ⇔ a = b;

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About a definition of metric over an abelian linearly ordered group

3. d(a, b) = d(b, a);

4. d(a, b) ≤ d(a, c)⊙d(b, c).

Proposition 3.2. [7] Let G = (G,⊙,≤) be an alo-group. Then, the operation

dG : (a, b) ∈ G2 → dG(a, b) = ||a÷ b|| ∈ G (11)

is a G-distance.

Proposition 3.3. [7] Let G = (G,⊙,≤) and G′ = (G′,◦,≤) be alo-groups,
h : G → G′ an isomorphism between G and G′. Then, for each choice of
a, b ∈ G :

dG′(h(a), h(b)) = h(dG(a, b)). (12)

Corolary 3.1. Let h : G → G′ be an isomorphism between the alo-groups
G = (G,⊙,≤) and G′ = (G′,◦,≤). If a′ = h(a), b′ = h(b), r′ = h(r) ∈ G′,
then r > e ⇔ r′ > e′ and

dG′(a
′, b′) < r′ ⇔ dG(a, b) < r.

4 Examples of continuous alo-groups over a

real interval

An alo-group G = (G,⊙,≤) is a real alo-group if and only if G is a subset
of the real line R and ≤ is the total order on G inherited from the usual
order on R. If G is a proper interval of R then, by Proposition 2.1, it is an
open interval.

Examples of real divisible continuous alo-groups are the following (see [8]
[9]):

Additive alo-group R = (R, +,≤), where + is the usual addition on R.
Then, e = 0 and for a, b ∈ R and n ∈ N:

a(−1) = −a, a÷ b = a− b, a(n) = na.

The norm ||a|| = |a| = a∨ (−a) generates the usual distance over R:

dR(a, b) = |a− b| = (a− b)∨ (b−a).

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B. Cavallo and L. D’Apuzzo

Multiplicative alo-group ]0,+∞[ = (]0, +∞[, ·,≤), where · is the usual
multiplication on R. Then, e = 1 and for a, b ∈]0, +∞[ and n ∈ N:

a(−1) = 1/a, a÷ b =
a

b
, a(n) = an.

The norm ||a|| = |a| = a∨a−1 generates the following ]0,+∞[ - distance

d]0,+∞[(a, b) =
a

b
∨

b

a
.

Fuzzy alo-group ]0,1[= (]0, 1[,⊗,≤), where ⊗ is the binary operation in
]0, 1[:

⊗ : (a, b) ∈]0, 1[×]0, 1[ 7→
ab

ab + (1−a)(1− b)
∈]0, 1[, (13)

Then, 0.5 is the identity element, 1 − a is the inverse of a ∈]0, 1[,
a÷ b = a(1−b)

a(1−b)+(1−a)b, a
(0) = 0.5,

a(n) =
an

an + (1−a)n
∀n ∈ N (14)

and

d]0,1[(a, b) =
a(1− b)

a(1− b) + (1−a)b
∨

b(1−a)
b(1−a) + (1− b)a

=
a(1− b)∨ b(1−a)
a(1− b) + b(1−a)

.

(15)

Remark 4.1. By Proposition 2.1, the closed unit interval [0, 1] can not be
structured as an alo-group; thus, in [7], the authors propose ⊗ as a suitable
binary operation on ]0, 1[, satisfying the following requirements: 0.5 is the
identity element with respect to ⊗; 1 − a is the inverse of a ∈]0, 1[ with
respect to ⊗; (]0, 1[,⊗,≤) is an alo-group. The operation ⊗ is the restriction
to ]0, 1[×]0, 1[ of the uninorm:

U(a, b) =

{
0, (a, b) ∈{(0, 1), (1, 0)};

ab
ab+(1−a)(1−b), otherwise.

The uninorms have been introduced in [12] as a generalization of t-norm
and t-conorm [13] and are commutative and associative operations on [0, 1],
verifying the monotonicity property (1).

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About a definition of metric over an abelian linearly ordered group

5 dG- neighborhoods and order topology

In this section G = (G,⊙,≤) is an alo-group and dG the G-distance in
(11).

Definition 5.1. Let c, r ∈ G and r > e; then the dG-neighborhood of c with
radius r is the set:

NdG(c; r) = {x ∈ G : dG(x, c) < r}. (16)

Of course c ∈ NdG(c; r) for each r > e. Then, NdG(c) will denote a dG-
neighborhood of c and NdG the set of the all dG-neighborhoods of the elements
of G.

Proposition 5.1. Let c, r ∈ G and r > e; then:

NdG(c; r) =]c÷ r, c⊙ r[

Proof. By properties (2), (3), (4) we get c ÷ r = c ⊙ r(−1) < c < c ⊙ r
and:

x ∈ NdG(c; r)

⇕


e ≤ x÷ c < r
or

e < c÷x < r

⇕


e ≤ x÷ c < r
or

r(−1) < x÷ c < e

⇕


c ≤ x < c⊙ r
or

c÷ r < x < c

⇕

x ∈]c÷ r, c⊙ r[.

2

9



B. Cavallo and L. D’Apuzzo

Proposition 5.2. Let h : G → G′ be an isomorphism between the alo-group
G = (G,⊙,≤) and the alo-group G′ = (G′,◦,≤). Then, for each choice of
c, r ∈ G and c′, r′ ∈ G such that c′ = h(c), r > e and r′ = h(r), the following
equality holds:

NdG′ (c
′; r′) = h(NdG(c; r)). (17)

Proof. By Proposition 3.3 and Corollary 3.1. 2

Example 5.1. The neighborhoods related to the examples in Section 4 are
the following:

• in the additive alo-group R = (R, +,≤), the neighborhood of c with
radius r is the open interval ]c− r, c + r[;

• in the multiplicative alo-group ]0,+∞[ = (]0, +∞[, ·,≤), the neighbor-
hood of c with radius r is the interval ]c

r
, c · r[;

• in the fuzzy alo-group ]0,1[= (]0, 1[,⊗,≤), the neighborhood of c with
radius r is the open interval ]

c(1−r)
c(1−r)+(1−c)r,

cr
cr+(1−c)(1−r)[.

By Proposition 5.1, NdG(c; r) is a particular neighborhood of c in the
order topology τG. We show by means of the following results that the set
NdG generates the order topology associated to G.

Proposition 5.3. Let A be an open set in the order topology τG. Then for
each c ∈ A there exists a dG-neighborhood of c included in A.

Proof. It is enough to prove the assertion in the case that A is an open
interval ]a, b[. Let c ∈]a, b[ and r = dG(a, c)∧dG(b, c) = (c÷a)∧ (b÷c). Let
us consider the cases:

1. r = c÷a ≤ b÷ c;

2. r = b÷ c < c÷a.

In the first case, a = c ÷ r, c ⊙ r ≤ b and so ]c ÷ r, c ⊙ r[⊆ A =]a, b[;
thus, by Proposition 5.1, NdG(c; r) ⊆ A. In the second case, the inclusion
NdG(c; r) ⊆ A can be proved by similar arguments. 2

Corolary 5.1. The set NdG of the all dG-neighborhoods of the elements of G
is a base for the order topology τG.

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About a definition of metric over an abelian linearly ordered group

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