Clonoids

Description

While the functions in a clone are closed under arbitrary compositions, a number of weaker closure properties have beenstudied by many authors including, e.g., Couceiro, Foldes, Harnau, Lehtonen, and Pippenger. In 2014, P.\ Mayr and the author introduced \emph{clonoids}; a clonoid is a set of finitary functions from a set $A$ into an algebra $\mathbf{B}$ that is closed under taking minors, and under the basic operations of $\mathbf{B}$. The proofs of the following results use clonoids: Every subvariety of a finitely generated variety with cube term is finitely generated (Aichinger, Mayr 2014). There are infinitely many not finitely generated clones on $\mathbb{Z}_p \times \mathbb{Z}_p$ containing $+$ (Kreinecker 2020). A finite abelian group has finitely many term-inequivalent expansions if and only if it is of squarefree order (Fioravanti 2020). In addition, clonoids have provided a new proof of a theorem by A.\ Pinus that on a finite set there are only finitely many algebraic geometries that are closed under union (Aichinger, Rossi, Sparks 2020). We will discuss these results and state open problems that involve clones and clonoids.

Period	11 Jun 2021
Event title	BLAST 2021
Event type	Conference
Location	AustriaShow on map