Algebraic Collaborations Workshop at the algebra seminar, April 4, 11, 13 and 25

The Mathematics Department dedicates four installments of the Algebra Seminar to a series of talks devoted to a locally recently developed notion: operation collaborations. A set S with two operations, * and o, is endowed with a new operation # which is a collaboration between and o in the sense that for all a, b in S, a#b is either a​*b or aob.

These Algebraic Collaborations Workshops are on April 4, 11, 13 and 25 from 4-5 p.m. via Zoom.

On April 25, from 4-5 p.m., the speaker is Asiyeh Rafieipour, Ohio University, discussing "Collaborations in the context of the magma monoid." Abstract: We revisit collaborations, the central topic of this workshop, and begin by returning to the two special examples that ignited the project: one-value and two-value graph magmas. We will revisit their relationship to graphs by seeing the exact mechanism by which they are induced from simple graphs. We then look at the characterizations of the associativity property on these magmas precisely in terms of such graphs. We continue by showing that the full sets of associative one-value and two-value graph magmas, as well as the collections of associative one-value and two-value graph magmas over a set S, are sub-semigroups of the Magma Monoid (M(S), $\triangleleft$). Motivated by these results, we investigate for what pairs of binary operations $\ast$ and $\circ$, the collaborations among those operations (a set we denote $<\ast,\circ>$ and to which we refer as {\it the collaborative span of} $\star$ and $\circ$) constitutes a sub-semigroup of (M(S), $\triangleleft$). A somewhat technical theorem achieves the goal and we will present it here it is still preliminary (but hopefully illuminating) state.

On April 13, from 4-5 p.m., the speaker is Aaron Nicely, Ohio University, discussing "Group collaborations and braces." Abstract: We explore the connections between the plus minus collaborations (also known as group collaborations) that yield groups (therefore called group group collaborations (with the repetition intended to be there) and the concept of (skew) left braces. Left Braces are an algebraic structure that generalizes Jacobson radical rings and are useful to find solutions to the Yang-Baxter equations.  This talk is the third one in the four-part collaborations workshop and the last one based on a manuscript by Lopez-Permouth, Nicely and Zailaee.

On April 11, from noon to 1 p.m., the speaker is Delfino Nolasco of Ohio University discussing "Pult-times collaborators" via Zoom. Abstract: (excerpt from the abstract of the paper in which the lecture is based): We consider Plus Times operations, collaborations on the two operations of the semiring of natural numbers. We produce an exhaustive list of associative collaborations between the usual addition and multiplication on the natural numbers $\mathbb{N}$. The Plus-Times operations we found are all examples of a type of construction that we call {\it augmentations by multidentities}. Understanding this fact simplifies in many cases the otherwise potentially cumbersome computations required to verify associativity. The basics on augmentations by multidentities are studied.

On April 4 the speaker is Thang Vo, of Ohio University discussing "Associativity of Plus-Minus collaborations." Abstract: We present the sections in a recent paper not-yet-published by Sergio López-Permouth, Aaron Nicely, and Majed Zailaee titled Semigroup collaborations between elementary operations. For this talk, we focus on the so called Plus-Minus collaborations, that is those operations which are collaborations between addition and subtraction on an Abelian group. We will start by considering Plus-Minus collaborations on the additive group of integers and then extend some of the results to arbitrary ordered groups.

About the Algebra Collaborations Workshop

Four of our doctoral students, Delfino Nolasco, Thang Vo, Aaron Nicely, and Asiyeh Rafieipour, will present talks which are either the result of their reading of previous work by others or results in their own dissertations. Specific titles and abstracts for the different talks will be announced.

Keeping with the spirit of a student seminar, their presentations will give some of them an opportunity to immerse themselves in a potential topic for their dissertation research and others a chance to share their own research results prior to defending them for their degree.