#### Presentation Title

Economical Generating Sets of the Monoid of Order-Preserving Partial Permutations POI(n)

#### Start Date

November 2016

#### End Date

November 2016

#### Location

HUB 302-4

#### Type of Presentation

Poster

#### Abstract

Semigroups and monoids are important in such fields as computer science, cryptography, abstract algebra and other branches of mathematics. In this research project we seek to determine economical generating sets for the monoid of order-preserving injections of an *n*-element set, POI(*n*). A generating set for any semigroup or monoid is a collection of elements *S* such that every element of the semigroup or monoid can be expressed as a product of elements from *S*. Generating sets are of fundamental importance in fields across mathematics and science, and mathematicians have great interest in studying generating sets of a variety of algebraic structures.

Using the algebra software package Groups, Algorithms, and Programming, GAP, we examine the key features of various generating sets of POI(*n*), including their sizes, the domains and ranges of the injections arising in them, and more. We proved that every generating set of POI(*n*) contains at least *n* elements, a fact that follows from a theorem we proved that characterizes generating sets of POI(*n*) according to the domains and ranges of its elements. Also, from our data we formulate a conjecture about the number of generating sets of POI(*n*).

Key words: Semigroups, monodies, POI(*n)*, generating sets, GAP.

Economical Generating Sets of the Monoid of Order-Preserving Partial Permutations POI(n)

HUB 302-4

Semigroups and monoids are important in such fields as computer science, cryptography, abstract algebra and other branches of mathematics. In this research project we seek to determine economical generating sets for the monoid of order-preserving injections of an *n*-element set, POI(*n*). A generating set for any semigroup or monoid is a collection of elements *S* such that every element of the semigroup or monoid can be expressed as a product of elements from *S*. Generating sets are of fundamental importance in fields across mathematics and science, and mathematicians have great interest in studying generating sets of a variety of algebraic structures.

Using the algebra software package Groups, Algorithms, and Programming, GAP, we examine the key features of various generating sets of POI(*n*), including their sizes, the domains and ranges of the injections arising in them, and more. We proved that every generating set of POI(*n*) contains at least *n* elements, a fact that follows from a theorem we proved that characterizes generating sets of POI(*n*) according to the domains and ranges of its elements. Also, from our data we formulate a conjecture about the number of generating sets of POI(*n*).

Key words: Semigroups, monodies, POI(*n)*, generating sets, GAP.