Let us try
to find out some more operations on permutations, but now assuming that
universal set is partitioned into subsets, namely:
.
If
permutation acts on , we write
,
if for all from to is injection from into so that images of induce partitioning on C that in general is distinct from .
Further we
are dealing with practical case when .
It is convenient
for us in place of universal set consider two
isomorphic nonintersecting sets and with bijection from to .
Thus, let be universal set so that , and is bijection from to ; with denoting . If permutation acts on [i.e., ], then we may wish sometimes to extend on ; so that this extension were . We would write for this new extended permutation
.
Let us
consider trivial extension with identity permutation on : .
Further,
for permutation we define other extension , so that for and and , there should hold [i.e., identity
permutation] and [i.e., isomorphically induced from toby ]:
It is easy
to see that holds
.
For
permutation we define twine permutation in order that holds
.
Thus, should be equal to . It is easy to see that . From here we get
.
Let us
prove technical lemma that helps to deal with permutations in some specific
cases: . Let and and , so that . The there holds
.
Proof
follows from direct calculation.
Let us use
new operations on permutations in order to calculate characteristics of maps.
First we
find how to calculate map’s image.
Permutations
act on . Corners that appear in image we attribute to set , that comes with bijection . Let be extension of
bijection, so that diagram commutes
Then for
permutation for which holds , we write
; .
This way
defined bijection coincide with that what we defined for image of p-map, if
only for twine permutation holds:
.
Indeed, we
defined twine permutation starting from expression
.
But, just
this form of definition of twine permutation were required in order it would
coincide with that what we defined by entering image of p-map. Indeed, if orbit
of is , then corresponding orbit of is , and corresponding orbit of is .