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3.3 Transformations on the Lattice
We apply the symmetry principle of the previous section to the lattice
L
. We will
refer to the subset
L
i
=
{
i
}×{
1
,
2
,...,
m
}
as the column
i
of
L
.
Let
F
n
,
m
=
F
repre-
=
{
,
,...,
}×{
,
,...,
}
sent the family of all subsets of
L
1
2
n
1
2
m
with just one point
{
,
,...,
}
{
,
,...,
}
in each column, that is, the set of all functions from
.
In the games we are interested in, the pure strategies for one of the players are ele-
ments of
1
2
n
to
1
2
m
F
m
satisfying different constraints, depending on the game. An
A
∈F
n
,
n
,
m
may be identified by the subset of
L
{
(
i
,
A
(
i
))
:
i
=
1
,
2
,...,
n
}
and it can also be rep-
resented simply by the vector
(
A
(
1
)
,
A
(
2
)
,...,
A
(
n
))
.Thus,
A
can be interpreted as
Fig. 3.2
Effect of transformations
T
i
over a subset
A
⊂
L
a
walk
along a linear set of
m
points at moments 1, 2, ...,
n
and also as a path
from the first to the last column of
L
which does not double back on itself. The
transformations
T
s
:
L
−→
L
,
s
=
1
,
2
,...,
n
defined by
⎧
⎨
(
i
,
j
)
,
if
i
=
s
,
T
s
(
i
,
j
)=
(
s
,
j
+
1
)
,
if
i
=
s
,
j
<
m
(3.3)
⎩
(
s
,
1
)
,
if
i
=
s
,
j
=
m
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