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 kth root? The classification of kth roots in a variety of mathematical structures has already proven to be useful, for instance, in fields such as cryptography. This research studies kth roots in the monoid of order-preserving partial permutations of an n-element set, denoted by POI(n). Our objective is to classify which elements of POI(n) have kth 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(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 kth roots. The data collection process included partitioning the elements of POI(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 kth roots for any n. Our current work aims to develop an all-encompassing theorem that characterizes, for any n, which elements of POI(n) have kth roots.

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Nov 23rd, 8:00 AM Nov 23rd, 8:45 AM

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 kth root? The classification of kth roots in a variety of mathematical structures has already proven to be useful, for instance, in fields such as cryptography. This research studies kth roots in the monoid of order-preserving partial permutations of an n-element set, denoted by POI(n). Our objective is to classify which elements of POI(n) have kth 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(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 kth roots. The data collection process included partitioning the elements of POI(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 kth roots for any n. Our current work aims to develop an all-encompassing theorem that characterizes, for any n, which elements of POI(n) have kth roots.