Vol 57, No 6 (2019)

This paper enters into a series of works on universal algebraic geometry—a branch of mathematics that is presently flourishing and is still undergoing active development. The theme and subject area of universal algebraic geometry have their origins in classical algebraic geometry over a field, while the language and almost the entire methodological apparatus belong to model theory and universal algebra. The focus of the paper is the problem of finding Dis-limits for a given algebraic structure \( \mathcal{A} \), i.e., algebraic structures in which all irreducible coordinate algebras over \( \mathcal{A} \) are embedded and in which there are no other finitely generated substructures. Finding a solution to this problem necessitated a good description of principal universal classes and quasivarieties. The paper is divided into two parts. In the first part, we give criteria for a given universal class (or quasivariety) to be principal. In the second part, we formulate explicitly the problem of finding Dis-limits for algebraic structures and show how the results of the first part make it possible to solve this problem in many cases.

We find a sufficient condition for a quasivariety K to have continuum many subquasivarieties that have no independent quasi-equational bases relative to K but have ω-independent quasi-equational bases relative to K. This condition also implies that K is Q-universal.

A group G is said to be rigid if it contains a normal series G = G₁ > G₂ > . . . > G_m > G_m+1 = 1, whose quotients G_i/G_i+1 are Abelian and, treated as right ℤ[G/G_i]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient G_i/G_i+1 are divisible by nonzero elements of the ring ℤ[G/G_i]. Every rigid group is embedded in a divisible one. Our main result is the theorem which reads as follows. Let G be a divisible rigid group. Then the coincidence of ∃-types of same-length tuples of elements of the group G implies that these tuples are conjugate via an automorphism of G. As corollaries we state that divisible rigid groups are strongly ℵ₀-homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.