#### Presentation Title

On k^th Roots of Partial Order-preserving Injections

#### Faculty Mentor

Scott Annin

#### Start Date

23-11-2019 8:00 AM

#### End Date

23-11-2019 8:45 AM

#### Location

205

#### Session

poster 1

#### Type of Presentation

Poster

#### Subject Area

physical_mathematical_sciences

#### Abstract

Monoids are important in mathematics, appearing in a variety of areas within abstract algebra. They are algebraic structures with an associative binary operation. We can ask: given an integer *k* ≥ 2, does a given element of the monoid possess a *k ^{th}* root? The classification of

*k*roots in a variety of mathematical structures has already proven to be useful, for instance, in fields such as cryptography. This research studies

^{th}*k*roots in the monoid of order-preserving partial permutations of an

^{th}*n-*element set, denoted by POI(

*n*). Our objective is to classify which elements of POI(

*n*) have

*k*roots, as such an investigation is likely to have further applications in the field of number theory, and in particular, cryptography. This study is a natural next step in a line of inquiry that was recently conducted for the symmetric inverse monoid of all partial permutations, of which POI(

^{th}*n*) is a submonoid. We perform computations in POI(

*n*) using the software

*Groups, Algorithms, Programming (GAP).*In particular, for

*n ≤*7, we have determined precisely which elements of POI(

*n*) possess

*k*roots. The data collection process included partitioning the elements of POI(

^{th}*n*) into specific categories based on common attributes relating to our inquiry. Using this partition of elements, we develop partial results describing which elements of POI(

*n*) possess

*k*roots for any

^{th}*n*. Our current work aims to develop an all-encompassing theorem that characterizes, for any

*n,*which elements of POI(

*n*) have

*k*roots.

^{th}On k^th Roots of Partial Order-preserving Injections

205

Monoids are important in mathematics, appearing in a variety of areas within abstract algebra. They are algebraic structures with an associative binary operation. We can ask: given an integer *k* ≥ 2, does a given element of the monoid possess a *k ^{th}* root? The classification of

*k*roots in a variety of mathematical structures has already proven to be useful, for instance, in fields such as cryptography. This research studies

^{th}*k*roots in the monoid of order-preserving partial permutations of an

^{th}*n-*element set, denoted by POI(

*n*). Our objective is to classify which elements of POI(

*n*) have

*k*roots, as such an investigation is likely to have further applications in the field of number theory, and in particular, cryptography. This study is a natural next step in a line of inquiry that was recently conducted for the symmetric inverse monoid of all partial permutations, of which POI(

^{th}*n*) is a submonoid. We perform computations in POI(

*n*) using the software

*Groups, Algorithms, Programming (GAP).*In particular, for

*n ≤*7, we have determined precisely which elements of POI(

*n*) possess

*k*roots. The data collection process included partitioning the elements of POI(

^{th}*n*) into specific categories based on common attributes relating to our inquiry. Using this partition of elements, we develop partial results describing which elements of POI(

*n*) possess

*k*roots for any

^{th}*n*. Our current work aims to develop an all-encompassing theorem that characterizes, for any

*n,*which elements of POI(

*n*) have

*k*roots.

^{th}