Enumerating kth Roots in the Symmetric Inverse Monoid

Christopher W. York

Dr. Valentin V. Andreev, Mentor

Department of Mathematics

October 1, 2014

The Symmetric Inverse Monoid

• Denoted SIM(n), the symmetric inverse monoid is the set of all partial one-to-one mappings from the set {1,2,…,n} onto itself with the operation of composition

• For example, maps 1 to 5, 2 to itself, 3 to 1, 4 to 3, and 5 to nothing.

Cycle and Path Notation

• Every element in SIM(n) can be expressed as the product of disjoint paths and cycles

• Paths map a number to the one next to it and the last number to nothing and are denoted with brackets. For example, [12357] maps 1 to 2, 2 to 3, 3 to 5, 5 to 7, and 7 to nothing

• Cycles map the last number to the first number and are denoted with parenthesis. For example, (3452) maps 3 to 4, 4 to 5, 5 to 2, and 2 to 3

• Length of a path or cycle is the number of numbers in it. For example, [12357] is of length 5.

Raising Elements to a Power k

• Raising an element in SIM(n) to the th power means applying the mapping unto itself times, creating a “skipping by ” effect

• For example,

• This “breaks” a path into paths of lengths differing by at most 1

• Fact: Let where are disjoint paths and/or cycles. Then

Definition of kth Root

• An element is a kth root of if and only if .

• For example, in , [123456789] is a 4th root of [159][26][37][48].

• The aim is to find formulas to determine the number of kth roots any element of SIM(n).

Previous Research

• Annin et al. [2] first determined whether an element in the symmetric group, an algebraic structure similar to SIM(n), has a kth root

• Recently, Annin [1] determined whether an element in SIM(n) has a kth root

• both papers posed the question of how many kth roots an element has

Interlacing Paths

• Raising a path to the th power breaks it into paths, so creating a th root of an element would be the “interlacing” of paths in groups of .

• If all the paths can’t be legally interlaced, then there are no th roots of the element

• Ex.: The interlacing of [123], [45], [67], and [89] would be [146825793]. Clearly, .

• The order of paths in the interlacing matters

• There can only be paths starting with the longest paths and lengths varying by at most 1.

The Root Counting Function

• The number of distinct th roots an element will be denoted by .

• This is equivalent to the number of ways to interlace all the paths of in groups of .

A Simple Case

• Let where are disjoint paths of the same length all greater than 1.

• Then.

• There are interlacings, whose order doesn’t matter

• The order within the interlacings does matter

A slightly More Complex Case

• Let where are disjoint paths and the lengths of are equal and is of length 1 less than the others. All lengths are greater than 1.

• Then .

• Again, there are interlacings, whose order doesn’t matter

• The smaller path has to be at the end of the interlacing it’s in

• There is a probability that the smaller path will be in a right place

An element with two weakly varying lengths

• Let be the product of disjoint paths where the first paths are equal length and the other paths are of length 1 less those paths. All lengths are greater than 1.

• The general form of the number of roots is

• is the probability the smaller paths will be in the right places in the interlacings

Paths of length 1

• Fact: whenever .

• This is the only instance where raising a path to the th power “breaks” it into less than paths

• Therefore, proper interlacings of paths only length 1 can have any number of paths as long as it’s at most .

• For example, if and , proper interlacings of ’s paths can include [12], [1], [134], [1234].

• Partitions of the number of paths can represent this

Some Helpful Formulas

• Let where is a product of disjoint paths and is a product of disjoint cycles

• Then .

• Paths and cycles cannot be interlaced

• Let where and are products of disjoint paths such that all paths in are at longer than those in by at least 2.

• Then .

• Paths varying by lengths of more than one cannot be interlaced

Further Research

• Elements with more than two varying lengths

• Elements with cycles

• Elements with weakly varying lengths starting with paths length 1

• Creating programs to calculate the number roots

• Thank you for listening!

References

[1] Annin, S. et al., On k’th roots in the symmetric inverse monoid. Pi Mu Epsilon 13:6 (2012), 321-331.

[2] Annin, S., Jansen, T. and Smith, C., On k’th roots in the symmetric and alternating Groups, Pi Mu Epsilon Journal 12:10 (2009), 581-589.