On when the union of two algebraic sets is algebraic
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Universal algebraic geometry, first introduced by B. Plotkin, extends some basic notions of classical algebraic geometry to universal algebra.
For a clone C on a set A, a subset of An is called algebraic if it is the solution set of a system of equations from the n-ary part of C. A basic fact in classical algebraic geometry is that the union of two algebraic sets is algebraic. This is no longer true in the present setting, and clones with this special property are called equationally additive.
We prove that on a finite set with at least three elements there is a continuum number of equationally additive constantive clones. We characterize the Mal'cev algebras whose clone of polynomial functions is equationally additive in terms of properties of the binary term condition commutator, and equationally additive E-minimal algebras in terms of their TCT-type.
Joint work with E. Aichinger and M. Behrisch.