On when the union of two algebraic sets is algebraic

Description

For a ring $\mathbf{R}$, we call a subset $B$ of $R^n$ \emph{algebraic} if $B$ is the solution set of a system of polynomial equations. If $\mathbf{R}$ is an integral domain, then every finite union of algebraic sets is an algebraic set, and therefore, for each $n\in \mathbb{N}$, the algebraic subsets of $R^n$ are the closed sets of a topology on $R^n$. Algebraic structures with this property are called \emph{equationally additive}. We give a characterization of equationally additive algebras that have a Ma\v{l}cev polynomial, and of all equationally additive finite E-minimal algebras. Moreover, we prove that on a finite set with at least three elements there are exactly $2^{\aleph_0}$ equationally additive constantive clones.Joint work with E. Aichinger and M. Behrisch.

Period	07 Jan 2023
Event title	Joint Mathematics Meeting 2023
Event type	Conference
Location	United StatesShow on map