Universal Algebraic Geometry and Polynomial Interpolation

The main focus of this dissertation are the interpolation of partial functions and the
properties of solution sets of systems of equations in universal algebra.
Chapter 1 is a brief introduction to universal algebra. In Chapter 2 we discuss
the relation between clones, relational clones, strong partial clones, weak systems with
equality and universal algebraic geometry.
Chapter 3 focuses on the problem of interpolating partial operations defined on the
universe of an algebra A with polynomial operations of A. We provide a characterization
of strictly 1-affine complete congruence regular Mal’cev algebras that extends the
characterization given in [11] for finite expanded groups. Furthermore, we characterize
the finite Mal’cev algebras with the property that for a given k ∈ N, each quotient of A
is strictly k-polynomially rich. An algebra is strictly k-polynomially rich if each k-ary
partial type-preserving operation can be interpolated by a polynomial operation. Our
notion of partial type-preserving operation extends the classical one for total operations
as follows: For a Mal’cev algebra A, a total operation is type-preserving in our sense if
and only if it preserves the type of each prime quotient of A in the sense of [48].
Chapter 4 focuses on universal algebraic geometry and on the problem of characterizing those algebras A that share with fields the property that the union of two
algebraic sets is an algebraic set. Such algebras are called equational domains. We
build on [32] and produce a characterization of equational domains among E-minimal
algebras, algebras on the three-element set with a cyclic automorphism, and algebras in
a congruence permutable variety with a constantive clone of term operations.
In Chapter 5 we focus on the problem of establishing the number of algebras on
a finite set A with a distinct algebraic geometry, called in the literature algebraically
inequivalent. In [69] Pinus proved that there are only finitely many algebraically
inequivalent equational domains on a finite set A. In [80] Tóth and Waldhauser proved
that on the two-element set there are exactly 25 algebraically inequivalent algebras. We
show that on a set with at least three elements there are continuously many algebraically
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inequivalent algebras and countably many algebraically inequivalent Mal’cev algebras.