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We prove that if an ∀∃-formula is true on the inverse limit of the retractive spectrum of algebras, then it is also true on the direct limit and establish some consequences for functions definable on these limits.

In a previous paper, on a collection FA of functional clones on a set A, we introduced a natural metric d turning it into a topological (metric) space $$ {\mathfrak{F}}_A=\left\langle {F}_A;d\right\rangle . $$ In this paper, we describe the structure of neighborhoods of clones in spaces $$ {\mathfrak{F}}_A $$ and establish a number of consequences of this result.

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.

The paper is devoted to the questions of algebraic geometry of universal algebras, more precisely, to the structures of algebraic sets of this algebras. It is introduced the concept of broad universal algebra. Some natural examples of such universal algebras are given including the lattices of functional clones on sets, the groups of permutations on sets, the lattices of partitions on sets, the countable free Boolean algebras, and the direct powers of universal algebras and others. Some special features of the structures of algebraic sets of broad universal algebras are considered. We prove the algebraic $n$-completnes of broad universal algebras. The results on the structure of the quasiorder which is generated on the broad universal algebra by its inner homomorphisms (homomorphisms between subalgebras) are presented. Some estimations of the powers of algebraic sets of the broad universal algebras are given. Some results on the minimal sets which are generated of algebraic sets of the broad universal algebras are obtained.

The concept of an algebraic set is a basic concept of the classical algebraic geometry over fields. This concept, along with the concept of an algebraic lattice of algebraic sets is the basic concept of so-called algebraic geometry of universal algebras. Moreover, there are traditionally two approaches to the definition of algebraic sets: the first is a direct generalization of the classical situation of the concept of the algebraic set over a field and connected with the homomorphisms of free algebras in the considered algebra, the second is formulated within the framework of the traditional model theory. In this paper we propose another approach to the characterisation of algebraic sets based on the concept of the inner homomorphisms of some extensions of considered algebra. Based on this approach we introduce the other representation of the lattices of the algebraic sets of universal algebras. Also we propose the criterion, in terms of internal homomorphisms, of the coincidence of the families of the algebraic sets of universal algebras with identical basic sets.

The question about the structure of lattices of subclasses of various classes of algebras is one of the basic ones in universal algebra. The case under consideration most frequently concerns lattices of subvarieties (subquasivarieties) of varieties (quasivarieties) of universal algebras. A similar question is also meaningful for other classes of algebras, in particular, for universal (i.e., axiomatizable by ∀-formulas) classes of algebras. The union of two ∀-classes is itself a ∀-class, hence such lattices are distributive. As a rule, those lattices of subclasses are rather large and are not simply structured. In this connection, it is of interest to distinguish some sublattices of such lattices that would model certain properties of the lattices themselves. The present paper deals with a similar problem for ∀-classes and varieties of universal algebras.

We introduce the notions of n-algebraic complete algebras and n-algebraic completeness, which makes it possible to find algebraic closures of subsets of universal algebras. We clarify how the n-algebraic completeness of an algebra is connected with the pseudodirect product of algebras.

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.

In [1] we came up with some approach and problems associated with subsets of functional clones on a fixed set that consist of functions occurring in a clone with a fixed restriction on their arity. This approach receives further development.

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.

Two functional clones F and G on a set A are said to be algebraically equivalent if sets of solutions for F- and G-equations coincide on A. It is proved that pairwise algebraically nonequivalent existentially additive clones on finite sets A are finite in number. We come up with results on the structure of algebraic equivalence classes, including an equationally additive clone, in the lattices of all clones on finite sets.

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.

UDC 519.48 We establish a relationship between autostable and point-definable congruences on countable universal algebras. Bibliography :9 titles. The source of this paper is two-fold. First, since the main objects studied in a universal algebra are universal algebras up to an isomorphism, it is natural to require, working with derived objects (subalgebras, endomorphisms, automorphisms, congruences, and so on), that the derived objects (for isomorphic algebras) should be in a one-to-one correspondence independently of a particular isomorphism between the algebras. This leads to consideration of those derived objects on universal algebras that are stable under automorphisms of these algebras (in particular, autostable congruences). Second, it is natural to consider derived objects that are definable on the algebra in some language of logic L (L-definable congruences). In this paper, we study the main simplest properties of autostable and L-definable (for different languages L of logic) congruences and their interaction. A congruence θ on a universal algebra A = � A; σ� is autostable if for any a, b ∈ A and

We look into the relationship between the so-called additive universal algebras $$ \mathfrak{A} $$ 0= 〈A 0; σ 0〉 and $$ \mathfrak{A} $$ 1= 〈A 1; σ 1〉 with common universe A having the same algebraic sets (Alg n $$ \mathfrak{A} $$ 0 = Alg n $$ \mathfrak{A} $$ 1 for any n ∈ ω) and subalgebras (Sub $$ \mathfrak{A} $$ 0 = Sub $$ \mathfrak{A} $$ 1).

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.

Basic algebras form a common generalization of MV-algebras and of orthomodular lattices, the algebraic tool for axiomatization of many-valued Łukasiewicz logic and the logic of quantum mechanics. Hence, they are included among the socalled quantum structures. An important role is played by linearly ordered basic algebras because every subdirectly irreducible MV-algebra and every subdirectly irreducible commutative basic algebra is linearly ordered. Since subdirectly irreducible linearly ordered basic algebras exist of any infinite cardinality, the natural question is to describe all possible order types of these algebras. This problem is solved in the paper.

We propose criterions for the geometric completeness of a variety of algebras, i.e., for all non-one-element algebras of the variety to be geometrically complete.

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.

Research into universal algebras (for different classifications included) is generally confined to working with termal (or polynomial) functions of these algebras. Attempts to go beyond this scope while staying within the frames of functions naturally definable on the algebras under consideration led the author to the idea of studying conditional termal functions (and their different generalizations such as positively, elementarily conditional termal, implicit, and abstract functions). As a continuation of studies in naturally definable functions on universal algebras, we propose to consider L-definable functions, where L is some logical language. This most general approach turns out to be connected with a scheme for defining conditional termal functions and their generalizations, as well as with various derivative structures of universal algebras. Here we present L-definable functions on universal algebras and outline their basic properties. On this basis, also, we introduce the notion of L-definably equivalent algebras, which is a generalization of the concept of being rationally equivalent for algebras.

We study the Galois correspondence between subgroups of groups of universal algebras automorphisms and subalgebras of fixed points of these automorphisms.

We propose a criterion for a subsemigroup of the semigroup of inner homomorphisms of a universal algebra to be the semigroup of inner homomorphisms of some enrichment of this algebra. Bibliography: 9 titles.

Some examples are given of applications of an in nite logical language in universal algebra, in the algebraic geometry of universal algebras, in the theory of implicit operations on algebras, in the Galois theory between automorphisms of universal algebras and its subalgebras of xed points, in the theory of Hamiltonian closure of subalgebras and other areas.

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 basis for classifying universal algebras is conventionally formed by one or other type of equivalence relations between the algebras, such as isomorphism, elementary equivalence, equivalence of algebras in different logical languages, geometric equivalence, and so on. Of crucial importance in this event are results that reduce one of such equivalences to another. The most important example (enjoying multiple applications) is the theorem stating that every two algebras are elementarily equivalent iff their ultrapowers with respect to some ultrafilters are isomorphic. Similar results are established for various equivalences of algebras related to algebraic geometry of universal algebras.

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.

Using the congruences of free algebras as well as the concepts of a conditional term and an implicit operation, a unifying method for studying algebraic and logically definable subsets of universal algebras is suggested. An overview of the results of the author in this field of research is included.

In studying derived objects on universal algebras, such as automorphisms, endomorphisms, congruences, subalgebras, etc., we are naturally interested in those that can be defined by the means of the universal algebras themselves (i.e., are definable in one sense or another)—in particular, in which part of all relevant derived objects is constituted by these. It is proved that for any algebraic lattice L and any of its 0-1-lower subsemilattices L 0 ⊆ L 1 ⊆ L 2, there exist a universal algebra $ \mathcal{A} $ and an isomorphism φ of the lattice L onto the lattice Sub $ \mathcal{A} $ such that φ(L 0) = OFSub $ \mathcal{A} $ , φ(L 1) = POFSub $ \mathcal{A} $ , φ(L 2) = FSub $ \mathcal{A} $ , and PFSub $ \mathcal{A} $ = FSub $ \mathcal{A} $ .

Using the notion of an implicit operation on universal algebras, we redefine basic notions of the algebraic geometry of universal algebras. The results obtained for the implicit algebraic geometry imply (as special cases) the known results on the conditional geometric and algebraic geometric comparability of algebras.

We consider problems of comparing universal algebras in respect of their conditional algebraic geometries. Such comparisons admit of a quite natural algebraic interpretation. Geometric scales for varieties of algebras constructed based on these relations are a natural tool for classifying the varieties of algebras, discriminator varieties in particular.

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

We introduce a preorder for universal algebras with respect to their geometries. This naturally leads to the notion of the geometric scale for a variety of algebras. We investigate connections between the introduced relation and infinite quasi-identities that hold in algebras, as well as other properties of the relation and the scale.

The concept of the scale of local computability (local program-computability possibilities) of all universal algebras is introduced. The properties of this scale are studied.

We present a series of concepts and prove some results on implicit operations on the various categories of universal algebras. This generalizes the previous results for pseudovarieties, pseudouniversal classes of algebras, positively conditional pseudovarieties, and so on.

Definable subsets are distinguished, based on which automorphisms of the scale of all finite algebras are studied.

We find an elementary equivalence criterion for the lattices of subalgebras of free algebras in regular varieties. The question is addressed of elementary equivalence for the automorphism groups of algebras of this type.

Comparison is suggested between the computability potential scales of algebras of different cardinalities and prove the undecidability of the elementary theory of the computability potential scale of all finite algebras.

We prove some representation theorems for lattices and their lower subsemilattices as the lattices of congruences and subsemilattices of first-order definable congruences of universal algebras.

We give a review of general known results on problems concerning the structure of scales of computability potentials of n-element algebras. We also formulate a number of natural open problems.

We study the syntactic descriptions of implicit operations on positively conditional pseudovarieties of finite algebras and the questions of axiomatizability of positively conditional pseudovarieties.