A Study to Generate a Weak Order from a Partially Ordered Set, Taken. . .
65
Table 1 data of 11 metals and their scores in lichens in 10 sites
metals
Al
As
Cd
Cu
Fe
Hg
Mn
Ni
Pb
V
Zn
Sites
st1
st2
st3
st4
st5
st6
st7
st8
st9
st10
data matrix
st1
st2
st3
st4
st5
st6
st7
st8
st9
st10
Al
4
5
7
5
5
4
4
4
3
5
As
2
5
7
6
7
5
5
4
4
4
Cd
5
7
7
7
4
3
4
2
2
3
Cu
5
5
7
7
4
3
5
3
3
4
Fe
7
7
7
7
7
7
7
7
6
7
Hg
4
3
5
3
3
3
4
2
2
2
Mn
7
7
7
7
6
6
7
4
5
6
Ni
7
7
7
7
6
5
7
4
4
7
Pb
7
7
7
7
5
4
6
4
7
4
V
7
7
7
7
7
6
7
7
4
7
Zn
7
7
7
7
7
7
7
6
7
7
term) is highly concentrated. As sites we select only 10 (of 20) because of reasons
which become clear in following sections. The concentrations are transformed into
a scale of integers from 1 to 7, following the suggestion of Nimis and Bargagli
(1999). 1 indicates a high naturality, whereas 7 express a high deviation from the
natural state. The data are shown in Table 1, in the Results-section. An entry of the
data matrix, dm(I,j) is associated with jth site and the ith metalloid. Details can be
found in Pirintsos et al. (2014).
2.2 Basic Concepts of Partial Order
Let X be a finite set of n objects, labeled by x(i) (i = 1, . . . ,n). Objects could be but
not limited to
• chemical compounds (here: metals/metalloids)
• nations, characterized by for example child well-being indicators
• strategies, characterized by performance indicators
• geographical units, characterized for example by pollution, or (within a socioeconomic context) by poverty indicators
Here, indeed the elements of the real example are “chemical elements”, namely
n = 11 metals.
To define an order relation among them, the relation “≤” has to obey the
following order axioms:
65
Table 1 data of 11 metals and their scores in lichens in 10 sites
metals
Al
As
Cd
Cu
Fe
Hg
Mn
Ni
Pb
V
Zn
Sites
st1
st2
st3
st4
st5
st6
st7
st8
st9
st10
data matrix
st1
st2
st3
st4
st5
st6
st7
st8
st9
st10
Al
4
5
7
5
5
4
4
4
3
5
As
2
5
7
6
7
5
5
4
4
4
Cd
5
7
7
7
4
3
4
2
2
3
Cu
5
5
7
7
4
3
5
3
3
4
Fe
7
7
7
7
7
7
7
7
6
7
Hg
4
3
5
3
3
3
4
2
2
2
Mn
7
7
7
7
6
6
7
4
5
6
Ni
7
7
7
7
6
5
7
4
4
7
Pb
7
7
7
7
5
4
6
4
7
4
V
7
7
7
7
7
6
7
7
4
7
Zn
7
7
7
7
7
7
7
6
7
7
term) is highly concentrated. As sites we select only 10 (of 20) because of reasons
which become clear in following sections. The concentrations are transformed into
a scale of integers from 1 to 7, following the suggestion of Nimis and Bargagli
(1999). 1 indicates a high naturality, whereas 7 express a high deviation from the
natural state. The data are shown in Table 1, in the Results-section. An entry of the
data matrix, dm(I,j) is associated with jth site and the ith metalloid. Details can be
found in Pirintsos et al. (2014).
2.2 Basic Concepts of Partial Order
Let X be a finite set of n objects, labeled by x(i) (i = 1, . . . ,n). Objects could be but
not limited to
• chemical compounds (here: metals/metalloids)
• nations, characterized by for example child well-being indicators
• strategies, characterized by performance indicators
• geographical units, characterized for example by pollution, or (within a socioeconomic context) by poverty indicators
Here, indeed the elements of the real example are “chemical elements”, namely
n = 11 metals.
To define an order relation among them, the relation “≤” has to obey the
following order axioms: