More about permutations

 

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.

 

 

Calculation of image of partial map and its characteristics

 

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 .