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.
Sprache der Kurzfassung:
Englisch
Vortragstyp:
Hauptvortrag / Eingeladener Vortrag auf einer Tagung