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Basing on the notion of algebraic function in real function theory, we study the question of the syntactic definition of the functions on universal algebras whose graphs are algebraic or piecewise-algebraic sets.

We prove the universal equivalence of the direct and inverse limits of retractive spectra of universal algebras and give a few consequences of this assertion.

We study the equivalence relations between functional clones which are based on the coincidence of their quantifier-free formal subsets of the corresponding universal algebras.

The paper is a brief survey of the author’s results connected with the lattices of algebraic sets of universal algebras and with the operator of algebraic closure on the subsets of direct powers of basic sets of algebras.

We consider the existence problems for quasiorders on sets in terms of which it is possible to describe the algebraic closure operator on subsets of universal algebras with a given universe.

We study the quasiorders on sets which are induced by inner homomorphisms of algebras with a given universe. These quasiorders play an important role on the study of the operator of algebraic closure on sets in universal algebras.

We introduce the concept of Hamiltonian closure of subalgebras of universal algebras and obtain a series of its properties and the properties of the lattices of Hamiltonian closed subalgebras of universal algebras.

The clones of functions on a set A are related to an arbitrary universal algebra $\mathfrak{A} = \left\langle {A;\sigma } \right\rangle $ in various natural ways. The simplest and minimal of them is the clone $Tr(\mathfrak{A})$ of termal functions of the algebra $\mathfrak{A}$ , that is, the closure of the collection of signature functions of these algebras with respect to the operator of superposition. The coincidence of similar clones (up to conjugation by the bijections of the underlying sets of the algebras) generates various equivalence relations on universal algebras, the first of which is the relation of rational equivalence introduced by Mal’tsev. This article deals with clone equivalences of this kind between algebras of arbitrary signatures.

The concept of implicit operation on pseudovarieties of semigroups goes back to Eilenberg and Schutzenberger [1]. The author in [2–5] generalized this concept to other classes of algebras and established a connection between these operations and positively conditional termal functions in the case of uniform local finiteness of the algebras of the class in question. In this article we put forth the concept of an implicit operation for an arbitrary universal algebra, not necessarily locally finite, and establish a connection between these operations and infinite analogs of positively conditional terms, as well as ∞-quasi-identities arising in the algebraic geometry of universal algebras. We also consider conditions for implicit equivalence of algebras to lattices, semilattices, and Boolean algebras.

We classify the discriminator varieties of universal algebras, basing on the structure of their conditional geometric scales.