# AAA 98 – Abstracts

## On Freese's amalgamation technique

Freese's amalgamation technique is a tool for finding certain lattices as sublattices of the congruence lattice of a given algebra. Namely if $\mathbf A$ is an algebra and $\alpha \in \operatorname{Con}(\mathbf A)$ we define and we look for sublattices in $\operatorname{Con}(\mathbf A^n(\alpha))$. Since $\mathbf A^n(\alpha)$ is a subalgebra of $\mathbf A^n$, all those sublattices are admissible in the variety generated by $\mathbf A$.

## Clones of polynomials

We present some properties of clones whose members are induced by polynomials in $D[\bar{x}]$ over some integral domain $D$. We use these polynomials to derive properties of finite nilpotent algebras in congruence modular varieties.

This is joint research with S. Kreinecker (JKU Linz) supported by the Austrian Science Fund (FWF):P29931.

## On the fraction of finite simple algebras

We calculate the ratio of algebras with a certain property $\Pi$. Define by $\operatorname{Alg}_{\rho,n}$ the set of algebras $\mathbf{A}$ of type $\rho$ with universe $\{1,\dots, n\}$ and let $\operatorname{Alg}_{\rho}=\bigcup_{n=1}^{\infty} \operatorname{Alg}_{\rho, n}$. Then the probability $P$ that $\mathbf{A}\in \operatorname{Alg}_{\rho}$ has property $\Pi$ has been defined by where

It is a well known fact that in this sense the probability that an algebra has no trivial subalgebras is 1. In 1975, V.L. Murskii proved that almost every algebra is idemprimal and even primal with some restrictions on the signature. Hence, since every idemprimal algebra is simple, the fraction of simple algebras is $1$. However, the convergence rate can be improved a lot, using a more direct approach. We show that the fraction of simple $n$-element algebras is greater than $1-e^{-n^{k-1}+2n^{k-2}+n\text{ln}(n)}$ if there is at least one at least ternary function symbol in the signature.

## Module Cancellation Properties

All rings are commutative with identity and all modules are unitary. An $R$-module $M$ is called a cancellation (weak cancellation) module if for all ideals $I$ and $J$ of $R$, $IM=JM$ implies that $I=J$ ($I+\operatorname{ann}(M)=J+\operatorname{ann}(M)$). In this work we generalize these concepts to half-cancellation (half-weak cancellation) as follows. An $R$-module is a half- cancellation (half-weak cancellation) if for all ideals $I$ of $R$, $IM=M$ implies $I=R$ ($I+\operatorname{ann}(M)=R$). Also a submodule $N$ of an $R$-module $M$ is called half-joint sumodule of an $R$-module $M$ if for all ideals $I$ and all submodules $K$ of $M$, $N=IN+K$ implies that $R=I+[K:N]$. Several results of these submodules if M is a multiplication modules are considered. We also give a particular attention to the idealization of these modules. In particular we study if $I+N$ is an ideal of the ring $R(M)$ have a cancellation property the each of $I$ and $N$ have the same property. And under some conditions we see the converse of the above statement is true.

## A connection Between Multialgebras and Algebras Via fundamental Relation

In this note we use the fundamental relation to construct a functor between categories of multialgebras and algebras. Also, we investigate the behavior of some important objects such as product, coproduct and etc. under this functor. Finally, we introduced and study the free multialgebras and its behavior under fundamental functor.

## On ropheomorphisms and absorption classes

Based on the notion of absorption introduced by Barto and Kozik, we study the Galois correspondence between finitary operations and finitary relations on a (usually finite) set given by the binary relation of absorption. Namely, we say that an $n$-ary function absorbs an $m$-ary relation if for any $(m\times n)$-matrix whose columns belong to the relation with at most one exception, the $m$-tuple obtained by row-wise application of the function to the matrix belongs again to the relation.

We identify certain closure properties for the Galois closed sets of relations (absorption classes) and the closed sets of operations (sets of ropheomorphisms). Moreover, we show that there are infinitely many such classes for any carrier set of cardinality at least two. We conclude the talk with some comments on the Boolean case.

This presents joint work obtained together with Erkko Lehtonen, TU Dresden.

## Extending minion homomorphisms from the canonical subclone

Let $\mathfrak{A}$ be a finitely bounded homogeneous structure, and let $\mathfrak{B}$ be a finitely related reduct of $\mathfrak{A}$. It was conjectured by M. Bodirsky and M. Pinsker that in this case the CSP over the structure $\mathfrak{B}$ is either in $\mathbf{P}$ or $\mathbf{NP}$-complete. We know that if there exists a uniformly continuous minion homomorphism (minor preserving map) from the polymorphism clone of $\mathfrak{B}$ to the clone of projections, then $\operatorname{CSP}(\mathfrak{B})$ is $\mathbf{NP}$-complete (Barto, Opršal, Pinsker). On the other hand, we know that either there is such a minion homomorphism from the canonical subclone of $\operatorname{Pol}(\mathfrak{B})$, or $\mathfrak{B}$ has a canonical pseudo-cyclic polymorphism (Barto, Kozik, Pinsker). Moreover, in the latter case we know that $\operatorname{CSP}(\mathfrak{B})$ is tractable (Bulatov, Zhuk; Bodirsky, Mottet). This motivates the question whether we can extend a given minion homomorphism defined on the canonical subclone of $\operatorname{Pol}(\mathfrak{B})$ to a minion homomorphism from $\operatorname{Pol}(\mathfrak{B})$ to the clone of projections. During this talk I examine this question, and give some sufficient conditions which make sure that such an extension exists.

Using the tools presented in this talk, we managed to prove that CSP dichotomy holds for the class of finite coverings of unary $\omega$-categorical structures. My hope is that these tools can also be used to show the dichotomy conjecture for other classes of infinite structures, and thus get us closer proving the conjecture in its full generality.

Joint work with Manuel Bodirsky.

## Algebra for multivariate mappings over a finite set

We investigate multiple input output gates on a finite set. We investigate closure under serial and parallel composition, and demand that "wire permutations" are in our closed sets. We aim to use the tools similar to those of clone theory to investigate these closed sets. We are primarily interested in mappings that are bijections.

We will describe a number of results about the structure of closed sets. For instance we show that for odd $\vert A\vert$, there is a finite generatings set for all bijections, while for even $\vert A\vert$ there is no finite generating set.

## Multipermutations and positive first-order logic

We look at the monoid of surjective hyper operations, a.k.a. multipermutations, as it appeared in connection with study of the complexity of positive first order logic without equality. We relate it with the already studied subsemigroups of binary relations, give some of its structural properties and present an array of open questions.

## Relatively residuated lattices

It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplemntation cannot be converted in residuated ones. The aim of my talk is to introduce a more general concept of a relative residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relative residuated lattices and show that they form a variety which is arithmetic and congruence weakly regular. (co-authors: Helmut Länger, Jan Kühr)

## Three Concrete Complexity Questions about Constraints, in Search of Theories

An instance of the constraint satisfaction problem (CSP) consists of a set of constraints on a set of variables, and the goal is to decide if there exists an assignment to the variables satisfying all of the constraints. One obtains a rich family of problems by parameterizing by the so-called constraint language, a set of relations that may be used to pose constraints. The so-called algebraic approach to constraint satisfaction, whereby universal algebraic methods and techniques are used to gain insight into the complexity of such problems, has led to the establishment of comprehensive dichotomy theorems; these theorems completely describe the complexity of all problems within some large family (with respect to inclusion in some complexity class). Such theorems have been given for the vanilla CSP as well as for many different variations thereof.

In this talk, we describe three cases where there was a failure to establish a dichotomy theorem, and moreover, where (in each case) the failure can be evidenced by a single concrete computational problem that can be described simply (and apprehended by any undergraduate student!). For each case, we survey the background behind and the state-of-the-art of the problem, and also discuss how the problem points to the development of further and broader theory.

Parental advisory warning: this talk may include explicit content in the form of conjectures, speculation, and cash awards.

## Implicative weak BCK-algebras in Rickart rings

A Rickart ring is a ring $R$ such that for every $a\in R$ there exist idempotents $e,f\in R$ such that for all $x\in R$, $ax=0$ iff $ex=x$, and $xa=0$ iff $xf=x$. Strong Rickart rings are a subclass of Rickart rings which includes reduced Rickart rings and Rickart $*$-rings. It is known that on any strong Rickart ring, there is a version of the right star order (originating from $*$-rings of bounded linear Hilbert space operators).

Suppose $R$ is a strong Rickart ring $R$ which is a meet semilattice with respect to the right star order. Then $\langle R, *, 0 \rangle$, with $*$ defined by by $a*b = a - (a\wedge b)$, is an implicative weak BCK-algebra.

## Complexity of List Homomorphism Problem for Templates with Weak Jonsson Terms

We show that if a finite relational structure whose associated algebraic structure of polymorphisms has the property of being conservative; namely, that every subset is closed under all polymorphisms, and which omits types 1,2, and 5 in the sense of tame congruence theory, is in the complexity class NL. A consequence of this result is that the list homomorphism problem for binary templates with the aforementioned algebraic properties is either in L or is NL-complete.

## Characterizations and enumerations of classes of quasitrivial $n$-ary semigroups

Let $X$ be a nonempty set and let $n \geq 2$ be an integer. An $n$-ary operation $F\colon X^n \to X$ is said to be associative if it satisfies the following system of identities for all $x_1,\ldots,x_{2n-1}\in X$ and all $1\leq i\leq n-1$. In that case, the pair $(X,F)$ is called a $n$-ary semigroup. Recall also that an operation $F\colon X^n \to X$ is said to be reducible to a binary operation if it can be written as a composition of a binary associative operation. We are interested in $n$-ary associative operations that are quasitrivial (or conservative), i.e., operations that preserve unary relations: for every $x_1,\ldots,x_n\in X$, We will show that any associative and quasitrivial $n$-ary operation is reducible to a binary associative operation. We will also provide necessary and sufficient conditions under which the binary reduction is unique and quasitrivial. Finally, we will provide a characterization of these operations and, when the underlying set $X$ is finite, we will explicitly determine the sizes of classes of associative and quasitrivial operations that (i) contain at most one neutral element, (ii) contain exactly two neutral elements. As a by-product, these enumerations lead to several integer sequences that were previously unknown in the Sloane's On-Line Encyclopedia of Integer Sequences (e.g. A308351, A308352, and A308354).

## The classification of $R$-subgroups of the finite dimensional Beidleman near vector spaces.

Several researchers named Beidleman, Andre, Karzel and Whaling have introduced in different ways the theory of near-vector spaces. Our focus will be on the type of near vector spaces originally defined by Beidleman which uses the near-ring modules in the construction. In this talk we shall derive the finite dimensional Beidleman near-vector spaces and also present an algorithm that classifies its R-subgroups.

## Mal'cev conditions on the lattice of reflexive subalgebras of the square

We investigate Mal'cev conditions described by equations that are valid in the lattice of reflexive subalgebras of the square of an algebra. A reflexive subalgebra of the square of an algebra is a subalgebra of the square containing the diagonal. We give some examples of equations in the language $\{\vee, \circ \}$, characterizing when they are valid in the lattices of reflexive subalgebras of the square of algebras of a particular variety. To this end, we present a variant of the Pixley-Wille algorithm to compute sets of terms describing these properties.

This work was partially supported by the Austrian Science Fund FWF (P29931).

## Universal algebra meets actuarial science: Generalized metrics with applications to ratings and FCA

We introduce an order theoretic approach to generalized metrics that covers various concepts of distance. In particular, we point out the role of supermodular mappings on lattices and give a generalized framework for algebraic modeling of directed distances, which then can be applied in diverse settings such as comparison of ratings and formal concept lattices.

## Inconvertible posets and $n$-permutability

For a given poset, we can take an algebra in its polynomial clone, and then take an antisymmetric quasiorder of this algebra, thus gaining an other poset. This conversion induces a preorder on the class of posets. We examine this preorder and conclude that certain posets can only be converted to others that contain an isomorphic copy of them. As a consequence, we give two strong Mal'tsev-conditions that together imply $n$-permutability, but seperately do not.

## The Mathematics of Ivo Rosenberg

After a long struggle with Parkinson's disease, yet hard working and keeping a very kind spirit, Professor Ivo Rosenberg died on January 18, 2018.

Professor Ivo Rosenberg was an eminent scholar, brilliant mathematician, and one of the leading experts in universal algebra and discrete mathematics. His huge impact and contributions to these areas of mathematics are very hard to assess due to the great extension and ramifications of his works.

In this talk we will give a survey of some of the most influential works of Professor Ivo Rosenberg in mathematics.

## Cyclic codes over the matrix ring $M_3(\mathbb{F}_2)$

Let $M_3(\mathbb{F}_2)$ be the ring of all $3\times 3$ matrices over the field $\mathbb{F}_2$. In this article, we study the structures of cyclic codes over $M_3(\mathbb{F}_2)$. Also, we establish isometry between $M_{3}(\mathbb{F}_2)$ and $\mathbb{F}_{8}+u\mathbb{F}_{8}+u^2\mathbb{F}_{8}$, where $u^3=1$. As an application, we obtained several optimal codes over $\mathbb{F}_8$ from the Gray images of the cyclic codes over $M_{3}(\mathbb{F}_2)$.

## Cover problems, dimensions and the tensor product of complete lattices

This talk will treat different dimensional concepts of complete (ortho)lattices and their tensor products. The determination of these dimensions can be translated to certain set cover problems. This yields a sufficient condition for the multiplicativity of various lattice dimensions with respect to the tensor product of complete (ortho)lattices.

## Meet-irreducibles in $\mathcal{E}_A$

The system of all congruence lattices of all algebras defined on a fixed set $A$, ordered by inclusion, forms the lattice $\mathcal{E}_A$. Let $A$ be finite. We investigate the conditions under which the congruence lattice $\operatorname{Con}(A,f)$ for a connected monounary algebra $(A,f)$ is meet-irreducible.

## Counterexamples in finitely related structures

Two counterexamples relating to the dichotomy conjecture for CSPs of reducts of finitely bounded structures have been constructed recently. The first one is an $\omega$-categorical structure which satisfies non-trivial height 1 identities locally, but does not satisfy any such identities globally. The second counterexample is an $\omega$-categorical structure $\mathbb{A}$ such that the polymorphism clone of $\mathbb{A}$ has a non-continous clone homomorphism to the clone of projection. In the case of both counterexamples, however, the structures obtained have infinite signature. We use an encoding originally due to Hrushovski to show that the aforementioned counterexamples can be given as finitely related $\omega$-categorical structures.

This is a joint work with Pierre Gillibert, Michael Kompatscher, Antoine Mottet, and Michael Pinsker.

## Classifications of finitely generated semifields and lattice-ordered groups

I'll discuss several recent classification results concerning semifields. Using a theorem of Busaniche-Cabrer-Mundici on lattice-ordered groups, I'll describe all semifields that are finitely generated as semirings, and in particular show that they have to be additively idempotent (joint work with Miroslav Korbelar). Moreover, I'll sketch the study of Banach semifields by Eric Leichtnam and outline how it can (hopefully) be extended using the aforementioned results. All of these classifications are based on nice geometric tools which I'll explain in the talk.

## Circuits, equations and nilpotent algebras

The talk is intended to provide overview of the best known algorithms for solving equations in nilpotent algebras. Considered equations will be presented on input as circuits to compress input size. Both black-box and white-box approach will be presented and discussed.

## Weakening minor conditions

The purpose of this talk is to introduce a technique that allows one to take a nice minor condition and obtain a weaker minor condition that is still reasonably strong.

A minion $\mathcal M$ is a family of operations closed under taking minors (renaming variables, identifying variables, and adding dummy variables). A minor condition is a finite system of identities for operation symbols. Minor conditions are similar to Maltsev conditions for varieties of algebras. However, minor conditions never contain compositions because minions need not be closed under composition. (As a result, any minion with a constant operation satisfies all minor conditions.)

We will show how to produce, for a fixed positive integer $k$, a countable descending chain of weaker and weaker minor conditions $E_1,E_2,\dots$ such that whenever a minion $\mathcal M$ satisfies some $E_i$ then $\mathcal M$ contains a constant operation or an operation that depends on more than $k$ variables.

An example of a minor condition that can only be satisfied by either a constant operation or an operation that depends on at least 2 of its variables is \begin{align*} t(x,y,z,z)&\approx t(y,x,z,z)\\ t(z,z,x,y)&\approx t(z,z,y,x). \end{align*}

This is a joint work with Matthew Moore from the University of Kansas.

## Finite Relation Algebras and CSPs

We introduce the network satisfaction problem for a finite relational algebra and show its connection to the study of infinite domain constraint satisfaction problems.

## The equivalence problem for nilpotent algebras

The (circuit) equivalence problem of a finite algebra $\mathbf A$ is the computational problem where the input consists of two polynomials $p(x_1,\ldots,x_n)$ and $q(x_1,\ldots,x_n)$ (encoded by circuits), and the task is to decide whether $\mathbf A \models p(x_1,\ldots,x_n) \approx q(x_1,\ldots,x_n)$ or not.

For algebras from congruence modular varieties, the complexity of the equivalence problem is almost completely classified, essentially only leaving a gap for nilpotent, not supernilpotent algebras. In this talk I would like to show that the equivalence problem is in P for 2-nilpotent such algebras (joint work with Kawałek and Krzaczkowski) and discuss how the general nilpotent case relates to the equivalence problem for $\text{CC}^0$-circuits.

## Radical of filters in residuated lattices

We give a notion of radicals of filters in residuated lattices and prove some fundamental results about them, which are generalizations of those of BL-algebras. We show the characterization theorem of radicals of filters in residuated lattices. Moreover, we give an answer to the problem left open in A. B. Saeid and S. Zahiri, Radicals in MTL-algebras, Fuzzy Sets and Systems 236 (2014), 91–103.

## A Characterization of Concept Lattices of Relational Structures

A relational database can be represented by a relational structure $\Delta$. We consider conjunctive queries and SPJR algebra (a subset of relational algebra) as intensional and extensional viewpoints on the data, respectively. A Galois connection, one half of which is the result operation on queries, connects the intensional and extensional viewpoint, and unifies them through the notion of formal concepts (over $\Delta$). Thus, Formal Concept Analysis is merged with database theory. The Galois connection maps suprema of queries (given by conjunction / categorical sum) to infima of tables (given by the natural join), and vice versa; this was described in previous work. Further operations, like projection, can be captured by a (right) monoid action on concepts, so that we have three concept operations in total: supremum, infimum and outer multiplication. A characterization of concept lattices of relational structures in terms of these operations is presented.

## A certain lattice of quasivarieties

In the theory of natural dualities, it is a fundamental problem to determine which finite algebras have the property of being dualisable. A famous motivating example of a dualisable algebra is the two-element Boolean algebra, and the corresponding duality is known as Stone duality.

We have made significant progress in describing those finite aperiodic semigroups that are dualisable. Rather than detailing the dualisability problem, our talk will focus on a certain lattice that plays an important role in this description, and no knowledge of duality theory will be required.

We call a semigroup aperiodic if all of its subgroups are trivial. In our considerations, an important class of aperiodic semigroups is the variety generated by $\mathbf{R}$ and $\mathbf{P}$, where $\mathbf{R}$ is the two-element right-zero semigroup and $\mathbf{P}$ is a certain three-element semigroup. This can be described as the class of semigroups satisfying the equations $xxy \approx xy$ and $xyz \approx yxz$. Denote this class of semigroups by $\boldsymbol{\mathscr{V}}$.

As part of the classification of dualisable finite aperiodic semigroups, we are interested in describing the lattice, $\mathbf{L}$, of subquasivarieties of $\boldsymbol{\mathscr{V}}$. It was determined by Mark Sapir (around 40 years ago) that the lattice $\mathbf{L}$ is finite. However, the proof of a key lemma was omitted, and this lemma now appears to be incorrect (we have not yet determined whether the main results of Sapir's paper still hold). It turns out that $\mathbf{L}$ is indeed finite, but it is larger than was originally thought. We will give a full description of this lattice, and introduce a new four-element semigroup whose role was not previously known.

## Coordinatization of solvable algebras

A recent result by E. Aichinger states that we can expand a nilpotent algebra $\mathbf{A} = (A, F)$ in a congruence modular variety by a binary operation $+$, a unary operation $-$ and a nullary operation $0$ such that $(A,+,-,0)$ is an abelian group, and at least one central series of $\mathbf{A}$ is also a central series of $(A, F\cup \{+,-,0\})$. This allows to restrict questions about arbitrary nilpotent algebras to expanded groups. For nilpotent algebras of prime power order we can use finite fields to coordinatize the algebra, which can be used to bound the free spectrum and the complexity of solving equations. In this talk we will coordinatize certain solvable algebras.

Supported by the Austrian Science Fund (FWF): P29931.

## The Binary Comitant of a Menger Algebra of Term Rewriting Systems

A term rewriting system is the set of all rewrite rules, a method for transforming terms. It plays a major rule in many field of computer science, logic and mathematics. Using the concept of a Menger algebra of rank n which is a structure consisting a nonempty set together with the $(n + 1)$-ary operation satisfying the superassociative law, we define the $(n + 1)$-ary superposition operation on the set of all rewrite rules and so we can form a Menger algebra of term rewriting systems. The connection between semigroups and Menger algebras are very closely. This leads us to introduce the binary associative operation on this set called a diagonal semigroup and on its cartesian power called a binary comitant of Menger algebras of term rewriting systems. In this work we study some semigroup properties with these two binary operations and the isomorphism are given.

## Observables on quantum structures

We consider a question, for which quantum structures $E$, there is a one-to-one correspondence among observables on $E$ and spectral resolutions on $E$. Both these concepts are important in the field of quantum measurement. The bijection holds e.g. in the case $E$ is a $\sigma$-complete boolean algebra. The well-know Loomis-Sikorsky theorem takes an essential part in the proof of this. We have analogues of Loomis-Sikorsky theorem for $\sigma$-complete MV-algebras or monotone-sigma-complete effect algebras with Riesz Decomposition Property, and the bijection holds for these effect algebras as well. In the talk we show some other kinds of effect algebras when the bijection holds or fails, and we discuss the generalisation to higher dimensional observables.

## A study of generalized $k$-hypersubstitutions

In the study of linear hypersubstitutions leads us to extend this idea into the study of generalized linear hypersubstitutions. But there is a technical problem occurs in this defining. We try to avoid this problem to the study of generalized $k$-hypersubstitutions. In this talk, we present some of its interesting properties.

## On the structure of commutative nil clean rings

A ring is nil clean if any element can be represented as a sum of an idempotent and a nilpotent element. We survey some recent results about the structure of commutative nil clean rings and pose a couple of problems.

## Ordinal polynomials have finite big Ramsey degrees

In this talk we consider big Ramsey degrees of finite chains in certain countable ordinals. The first result in this direction is, of course, the infinite version of Ramsey's theorem which claims that finite chains all have the big Ramsey degree 1 in $\omega$. Another important step in this direction was Laver's result on divisibility of scattered chains which claims that the one-element chain has finite big Ramsey degree in any countable scattered chain. Scattered countable chains are of particular interest since the situation with non-scattered chains can be resolved fairly easily: non-scattered countable chains have the same finite big Ramsey degrees as the chain of rationals, where the big Ramsey degrees were explicitly computed by Devlin. Along the way we prove a result that we see as an infinite analogue of the Finite Product Ramsey Theorem.

## Generating subdirect products of lattices

It is quite straightforward that a direct product of two lattices (more generally idempotent algebras) is finitely generated if and only if both factors are finitely generated. However subdirect products of finitely generated lattices need not be finitely generated again. We investigate necessary and sufficient conditions for this to happen.

As particular subdirect products we consider fiber products (pullbacks) for epimorphisms $g\colon A\to D, h\colon B\to D$. Whether such a fiber product is finitely generated turns out to be intimately connected to whether $g$ and $h$ are bounded. Recall that a lattice epimorphism $g\colon A\to D$ is bounded if for any $d\in D$ the preimage $g^{-1}(d)$ is an interval in $A$. This is a classical notion due to McKenzie (1972) and Jonsson (1975). As a consequence of our results we obtain an apparently new characterization of bounded lattice homomorphisms in terms of their kernels.

This is join work with William DeMeo (Boulder, USA) and Nik Ruskuc (St Andrews, UK).

## Congruence Preserving Functions on Special $p$-Groups

For several classes of special $p$-groups, $G$, we address the question on when the near-ring $C_0(G)$ of congruence preserving functions is actually a ring.

## Finite degree clones are undecidable

A clone is a set of operations on a set that is closed under composition and variable manipulations. There are two common methods of finitely specifying a clone of operations. The first is to generate the clone from a finite set of operations via composition and variable manipulations. The second method is to specify the clone as all operations preserving a given finite set of relations. Clones specified in the first way are called finitely generated, and clones specified in the second way are called finitely related.

Since the 1970s, it has been conjectured that there is no algorithmic procedure for deciding whether a finitely generated clone is finitely related. We give an affirmative answer to this conjecture by associating to each Minsky machine $\mathcal{M}$ a finitely generated clone $\mathcal{C}$ such that $\mathcal{C}$ is finitely related if and only if $\mathcal{M}$ halts, thus proving that deciding whether a given clone is finitely related is impossible.

## Functorial Properties of the Reticulation of a Universal Algebra

The reticulation of an algebra $A$ from a variety ${\cal V}$ is a bounded distributive lattice ${\cal L}(A)$ whose spectrum of prime ideals (or filters) is homeomorphic to the spectrum of prime congruences of $A$, as topological spaces endowed with the Stone topologies. The reticulation is a very useful construction for transferring algebraic and topological properties between ${\cal V}$ and the variety ${\cal D}{\bf 01}$ of bounded distributive lattices, especially if a reticulation functor from ${\cal V}$ to ${\cal D}{\bf 01}$ can be defined. While a known property of bounded distributive lattices ensures the uniqueness up to a lattice isomorphism of the reticulation ${\cal L}(A)$ of $A$, the existence of ${\cal L}(A)$ has only been proven for several concrete varieties ${\cal V}$, out of which we mention unitary rings and commutative integral bounded residuated lattices. In [1], we have generalized all previous constructions of the reticulation for particular classes of algebras, by constructing the reticulation of $A$ in the case when the one-class congruence of $A$ is compact, the term condition commutator of $A$ is commutative and distributive w.r.t. arbitrary joins of congruences and the set ${\cal K}(A)$ of the compact congruences of $A$ is closed w.r.t. this commutator, in particular when ${\cal V}$ is congruence-modular and semi-degenerate and ${\cal K}(A)$ is closed w.r.t. the modular commutator, considering the prime congruences of $A$ w.r.t. the commutator operation. In the particular case when ${\cal V}$ is congruence-modular and semi-degenerate and all its members have their sets of compact congruences closed w.r.t. the modular commutator, we have also defined a reticulation functor ${\cal L}$ from the partial category of ${\cal V}$ whose morphisms are the surjective morphisms of ${\cal V}$ to ${\cal D}{\bf 01}$.

Now we extend the definition of ${\cal L}$ to all the morphisms of ${\cal V}$ that satisfy an additional condition we call the functoriality of the reticulation (FRet), which is fulfilled by all surjective morphisms, and study characterizations for FRet and varietal properties of ${\cal V}$ which ensure that all morphisms of ${\cal V}$ satisfy FRet, so that ${\cal L}$ is a functor from the whole variety ${\cal V}$ to ${\cal D}{\bf 01}$, along with some properties of this reticulation functor ${\cal L}$.

[1] G. Georgescu, C. Mureşan, The Reticulation of a Universal Algebra, Scientific Annals of Computer Science XXVIII, Issue 1 (2018), 67–113.

## Solving Systems of Equations over Certain Solvable Groups

We investigate the complexity of solving systems of polynomial equations over a finite group $G$. This problem is $\mathrm{NP}$-complete for non-abelian groups (Goldmann and Russell, 1999). Using the collecting procedure presented by Horváth and Szabó in 2006, we show that the problem becomes tractable for groups of order $|G|=pq$ for primes $p,q$ if we fix the number of equations.

## Classification of some special generalized derivations and endomorphisms of prime rings with involution

In my talk I will present some new classes of generalized derivations (endomorphisms) and study their connection with the structure of prime rings with involution of the second kind. Furthermore, we will provide examples to show that the various restrictions imposed in the hypotheses of our theorems are not superuous.

## Constructing large parametric families of quasigroups

A finite quasigroup is a pair $(Q, f)$ such that $Q$ is a finite set, $f\colon Q \times Q \to Q$ is a binary operation, and for any $a, b \in Q$ both equations $f (x, a) = b$ and $f (a, y) = b$ are (uniquely) solvable. Quasigroups are widely used in cryptographic applications, e.g., as substitution tables or components of round transformations of block ciphers. To secure against brute-force attacks, one should have large classes of quasigroups to choose from. Advantageous properties of quasigroups from the cryptographic standpoint are polynomial completeness and/or poor choice of subquasigroups.

We start with constructing parametric families of quasigroups of order $2^n$, and then present several generalizations of the construction. If $\vert Q \vert = 2^n$ for some $n \in \mathbb {N}$, then the elements $x$, $y$ of $Q$ can be encoded by binary $n$-tuples $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, and $f\colon (x,y)\to z=f(x,y)$, $z=(z_1,\ldots,z_n)$, can be thought of as a family $(f_1, \ldots, f_n)$, where $f_i$ are $2n$-ary Boolean functions.

A family $(g_1, \ldots, g_n)$ of $n$-ary Boolean functions is said to be proper (of order $n$) if for any two distinct binary $n$-tuples $(t_1, \ldots, t_n)$ and $(t'_1, \ldots, t'_n)$ there exists an index $i$, $1 \le i \le n$, such that $t_i \ne t'_i$, but $g_i (t_1, \ldots, t_n) = g_i (t'_1, \ldots, t'_n)$. Nosov proved that the functions $i=1, \ldots, n$, define a quasigroup operation for any choice of Boolean functions $\pi_1, \ldots, \pi_n$ if and only if the family $(g_1, \ldots, g_n)$ is proper (see e.g. [1]). Thus a single proper family of Boolean functions can identify up to $16^n$ quasigroups; however, the functions defined by different choices of $\pi_i$ may coincide.

This construction may be directly generalized to Cartesian products of arbitrary finite Abelian groups (see [2]) or products of quasigroups (in this case the operation is not necessarily associative and the expressions on the right-hand side of formulas (1) are evaluated, e.g., from left to right).

Another generalization consists in applying permutations to the components of the $n$-tuples $x$, $y$, and $z=f(x,y)$, where the three permutations, in general, may differ from each other ([3]).

This is a joint work with Alexey Galatenko.

#### Bibliography

[1] V. A. Nosov. Constructing Families of Latin Squares over Boolean Domains. Boolean Functions in Cryptology and Information Security, 2008, 200–207.

[2] V. A. Nosov, A. E. Pankratiev. Latin squares over abelian groups. Journal of Mathematical Sciences, 149, 3 (2008), 1230–1234.

[3] N. A. Piven. Investigation of quasigroups generated by proper families of Boolean functions of order 2. Intelligent Systems. Theory and Applications, 22, 1 (2018), 21–35 (in Russian).

## On polymorphism-homogeneous metric spaces with skeletons

Following a very fruitful structural property of homomorphism-homogeneity, it did not come as a surprise that a generalisation in the form of polymorphism-homogeneity was sooner or later bound to be introduced. A structure is endowed with the latter of these two properties in case every local polymorphism of it can be extended to a global polymorphism of the structure in point. Interestingly, the problem of deciding whether a metric space is homomorphism-homogeneous or not is known to be coNP-complete. For this reason, we focused our attention on the polymorphism-homogeneous metric spaces in our attempt to succeed in unveiling their exact structure. What came as a result was a fairly exciting approach triggered by the underlying graphs, the so-called skeletons, within a specific problematic class of metric spaces.

The aim of this talk is to present the most crucial results from an ongoing research on this topic. This is joint work with Maja Pech.

## On homomorphism- and polymorphism-homogeneous tournaments

The notions of homomorphism- and polymorphism-homogeneity are related to the notion of homogeneity from Fraïssé theory. While on the surface these three concepts look rather similar, their respective classification problems are quite different, each having its own beauties and difficulties. In this talk, we will report on our latest results concerning the classification of countable homomorphism- and polymorphism-homogeneous tournaments with loops allowed, generalizing previous results by Ilić, Mašulović, Rajković and Feller, who treated the finite case.

This is joint work with Christian Pech.

## Multisorted minions

In the paper The wonderland of reflections L. Barto, J. Opršal and M. Pinsker introduced so-called reflections of algebras as well as of clones. We look to these investigations from the point of view of multisorted algebras, which leads to multisorted minions (= minor closed sets of functions, also known as clonoids) and their reflections. Instead of relations one has to consider pairs of relations. Then reflections can be defined also for relational systems.

In the talk we present (together with all needed notions) the following theorem.

Let $\mathbb{A}=(A,Q_{A})$ and $\mathbb{B}=(B,Q_{B})$ be relational systems and let $\mathcal{M}_{1}=\operatorname{mPol} Q_{A}$, $\mathcal{M}_{2}=\operatorname{mPol} Q_{B}$ be the corresponding multisorted minions. Then the following are equivalent:

1. $\mathcal{M}_{2}\in\textsf{ERP}\mathcal{M}_{1}$,
2. There exists a minor-homomorphism $\lambda\colon \mathcal{M}_{1}\to\mathcal{M}_{2}$,
3. There exists $k\in\mathbb{N}_{+}$ and a relational system $\mathbb{B}_{1}=(A^{k},Q_{B_{1}})$ of the same signature as $\mathbb{B}$ such that
1. $\mathbb{B}_{1}$ is a reflection of $\mathbb{B}$
2. $\mathbb{B}_{1}$ is mc-constructible from $\mathbb{A}$, (i.e., $\forall (R,R')\in Q_{B_{1}}$ : $(\partial_{k}R,\partial_{k}R')\in [Q_{A}]_{\mathrm{mc}}$, where $[Q_{A}]_{\mathrm{mc}}$ is the so-called minor-closure of $Q_{A}$).

This is a joint work with E. Lehtonen.

## Skew constacyclic codes over $\mathbb{F}_q[u,v]/\langle u^2-1,v^2-v,uv-vu\rangle$

Let $\mathbb{F}_q$ be the Galois field of order $q=p^m$ where $p$ an odd prime and $m,$ a positive integer. In this article, we study the skew constacyclic codes over the finite non-chain ring $R=\mathbb{F}_q[u,v]/\langle u^2-1,v^2-v,uv-vu\rangle$. We define a Gray map and show that the Gray image of a skew constacyclic code of length $n$ over $R$ is a skew quasi-twisted code of index $4$ and length $4n$ over $\mathbb{F}_q$. Further, the necessary and sufficient condition of the skew constacyclic codes to be self-dual is obtained. Few non-trivial examples are presented to justify the derived results.

## A clonoid based approach to the logical equivalence of clones

In 2017 A. Pinus proved that for a finite set $A$, the number of pairwise algebraically non equivalent equationally additive clones is finite, as well as the number of pairwise logically inequivalent clones (w.r.t ${\sim}_{l_0 g}$-equivalence).

We put these two results in a more general frame and give an alternative proof of both using the Baker-Pixley-Sparks Lemma on clonoids (A. Sparks, 2018).

This is a joint work with E. Aichinger. Supported by the Austrian Science Fund (FWF), P29931.

## Measurement setups & a general approach to affine geometry

We will introduce the fundamental concept of measurement setup and, within this framework, we will discuss the impact on modeling general affine spaces.

During the talk we will shed light on the following guiding questions:

"What do we measure?"

"Wherin do we measure?"

"How do we measure?"

## Dynamics of the Hartman-Mycielski functor

A submeasure is a monotone, subadditive real-valued function on a Boolean algebra that sends the bottom element to zero. With every such submeasure one can associate a functor from the category of groups into the category of topological groups. This construction goes back to work of Hartman and Mycielski from the 1950s and has more recently attracted growing interest in both topological dynamics and model theory. The resulting topological groups have very peculiar properties, even for finite input groups. In my talk, I will explain this construction, survey some recent results, and discuss a very obvious, but interesting question: what happens if we replace groups by algebras?

## Residuated maps induced by poset and lattice-valued functions

The aim of the talk is to present some properties and applications of lattice-valued and generally of poset-valued functions in the framework of residuated maps.

If $L$ is a complete lattice and $X\neq\emptyset$, then a lattice valued function $\mu\colon X\rightarrow L$ induces a residuated map $f\colon L\rightarrow \mathcal{P}(X)$ whose values are the cuts of $\mu$, where $\mathcal{P}(X)$ is the power set of $X$ ordered dually to inclusion. Conversely, every residuated map $f$ from $L$ to the power set of $X$ determines a lattice valued function $\mu\colon X\rightarrow L$ whose cuts coincide with the values of $f$.

We give conditions under which the kernel of $f$ is a complete congruence on $L$. As a converse, for every complete congruence $\theta$ on $L$, there is an $L$-valued function $\mu$ with the co-domain $L$, which induces a residuated map, the kernel of which is $\theta$.

For finite lattices this procedure gives lattice representation theorems by meet-irreducible elements.

We also deal with poset-valued functions inducing residuated maps. We give conditions under which the map sending an element of a poset to the corresponding cut is quasi-residuated, and then conditions under which it is also residuated. We prove that without additional conditions, the map analogue to the residual is a partial function hence we get particular weakly residuated maps which, on the power set of the domain, generate centralized systems instead of closures. We show that main properties of residuated maps are preserved in this generalized case.

Further applications are related to $\Omega$-algebras which turn out to be determined by residuated maps induced by $\Omega$-valued equalities on the basic algebras, which are mappings from the square of an algebra to the complete lattice $\Omega$. In this case, the power set of the domain is replaced by the weak congruence lattice of the basic algebra. Representation theorems of $\Omega$-algebras are formulated in terms of residuated maps.

The presentation contains results obtained in collaboration with Eszter Horváth, Sándor Radeleczki and Andreja Tepavčević.

## On commutative BCK-algebras

Commutative BCK-algebras that can be embedded into Wajsberg algebras (MV-algebras) are called ŁBCK-algebras. We charecterize ŁBCK-algebras of finite height by means of forbidden subalgebras that are obtained from finite linearly ordered ŁBCK-algebras by duplicating the bottom element. Then we describe finitely generated covers of some notable varieties of commutative BCK-algebras.

## Computing some topological indices of fullerene linear and cyclic $(C_n)_m$ and symmetry group in $(C_n)_m$ fullerene

Topological indices are a numerical invariant of a chemical graph. In this paper, we apply the action of the symmetry group of a fullerenes molecule Cyclic and linear $(C_n)_m$ on the set of its chemical bonds to compute the PI and Szeged indices of $(C_n)_m$. This is an efficient method that can be applied for other classes of fullerenes.

## On the number of clonoids

A clonoid is a set of finitary functions from a set $A$ to a set $B$ that is closed under taking minors. Hence clonoids are generalizations of clones. Clonoids have recently been investigated in connection to classifying the complexity of Promise Constraint Satisfaction Problems. By a classical result of Post, there are only countably many clones on a $2$-element set. In contrast to that, we present continuum many clonoids for $A = B = \{0,1\}$. More generally, for any finite set $A$ and any $2$-element algebra $\mathbf{B}$, we give the cardinality of the set of clonoids from $A$ to $\mathbf{B}$ that are closed under the operations of $\mathbf{B}$.

## Quasigroups and the Yang-Baxter equation

I will explain how we got from a fundamental equation of mathematical physics to an intriguing combinatorial problem about latin squares.

## The return of the smooth digraphs modulo pp-constructability

We call a digraph smooth if every vertex has at least one incoming and one outgoing edge. We consider the set of all finite smooth digraphs ordered by pp-constructability, i.e., $A\geq B$ iff $A$ is pp-constructable from $B$. In a recent result Barto, Kozik, and Niven showed that the core of a smooth digraph with a WNU is a disjoint union of directed cycles. Therefore we will mostly talk about disjoint unions of cycles. In this talk we will present a complete (and nice) classification of this poset and show that every element (apart from the bottom element) is equivalent to a disjoint union of cycles whose lengths are square-free.

## Clones of random algebras satisfying idempotent linear Maltsev conditions

Let $L$ be a finite algebraic language with at least one operation symbol of arity $>1$. By a result of Murskii (1975), a random finite $L$-algebra is almost surely a semiprimal algebra with no proper subalgebras of size $>1$. In a recent joint paper with Cliff Bergman (2018+) we looked at the analogous problem when the probability space is restricted to the class of all finite models of a set $\Sigma$ of idempotent linear $L$-identities, i.e., the identities of a strong, idempotent, linear Maltsev condition. We found a simple syntactic condition $(*)$ such that $\Sigma$ satisfies $(*)$ if and only if a random finite model of $\Sigma$ is almost surely idemprimal.

In the talk I will discuss the following question: Which clones occur with positive probability among the clones of random finite models of $\Sigma$? Clearly, this question is interesting only if $(*)$ fails for $\Sigma$; this is the case, for example, if $\Sigma$ is the set of identities for a Maltsev term, or majority term, or minority term, or semiprojection term.

## Extending maps to profinite completions in the variety of median algebras

Recall that a ternary algebra $(A, \mathbf{m})$ is a median algebra if it satisfies the equational theory of the median term $m(x,y,z):=(x\vee y) \wedge (x\vee z) \wedge (y \vee z)$ in the variety of bounded distributive lattices. Every median algebra $\mathbf{A}$ is residually finite, and has a profinite completion $\hat{\mathbf{A}}$. In AAA97, a method to construct $\hat{\mathbf{A}}$ based on natural dualities was exposed, and order-theoretic properties of $\hat{\mathbf{A}}$ were investigated.

In this talk, we go a step further and investigate two related problems:

1. Given two median algebras $\mathbf{A}$, $\mathbf{B}$ and a map $u\colon \mathbf{A} \to \mathbf{B}$, how to define a `reasonable' extension $u^\delta \colon \hat{\mathbf{A}} \to \hat{\mathbf{B}}$ of $u$?
2. How to investigate preservation of equations of median algebra expansions to their profinite completions?

Both problems can be dealt with the introduction of a suitable topology on profinite completions of median algebras. We illustrate the constructions by considering the case of median algebras with a retraction.

## IBN-varieties of algebras

One of the important question of the universal algebraic geometry is a question about difference between geometric and automorphic equivalence of algebras of the some variety $\Theta$. If the variety $\Theta$ is an IBN-variety (has an IBN propriety) then the studying of this question is easier.

Will be presented the very simple proof of the very strong

Theorem. If $\Delta$ is an IBN-variety of algebras then every it supvariety $\Theta \supset \Delta$ is also an IBN-variety.

We will consider many varieties of one-sorted and many-sorted algebras, which appears in the algebraic studies by very natural way. By using of this Theorem we will prove that these varieties have an IBN propriety.

## Congruence preserving operations on rings $Z_n$

We investigate clones $\operatorname{Comp}(Z_n)$ of compatible (congruence preserving) functions on ring $Z_n$. It is known that this clone coincides with the clone $P(Z_n)$ of polynomial functions iff $n$ is square-free. We investigate clones between $\operatorname{Comp}(Z_n)$ and $P(Z_n)$ for $n=p^2$ and $n=p^3$, where $p$ is prime.

## Congruences and subdirect representations of graphs

A congruence is defined on a graph (graph here means non-directed graph with single edges and a vertex may have a loop). It is shown that the well-known isomorphism theorems have analogues for graphs. Moreover, as in algebra, subdirect representations of graphs can be expressed in terms of congruences. It is also shown that every non-trivial graph is a subdirect product of subdirectly irreducuble graphs; the latter graphs are explicitly determined.

## The power of linear programming for infinite-domain valued CSPs

Given a set $D$ (the domain), a valued constraint language over $D$ is a set a finite set of functions $f$, also called cost functions, defined on (a subset of) $D$ and having rational values.  Given a valued constraint language $\Gamma$, the valued constraint satisfaction problem for $\Gamma$, $\operatorname{VCSP}(\Gamma)$, takes as input a set of variables $V$, a finite sum $\phi$ of cost functions from $\Gamma$ applied to the variables in $V$, and a rational number $u$; the computational task is to decide whether there exists an assignment of values from $D$ for the variables in $V$ such that the cost of the assignment is not greater than $u$. If $D$ is finite then solving $\operatorname{VCSP}(\Gamma)$ for a given instance is computationally equivalent to finding the assignment for the variables with minimum cost. The computational complexity of the VCSP for languages over a finite domain has been studied extensively [Kolmogorov-Krokhin-Rolínek, 2015 + Bulatov-Zhuk, 2017] Every finite-domain VCSP has a natural Linear Programming relaxation, called the basic linear programming relaxation, BLP, and a milestone in the study of finite-domain VCSP is the result of Kolmogorov, Thapper, and Živný [V. Kolmogorov, J. Thapper, and S. Živný. The power of linear programming for general-valued CSPs. SIAM J. Comput., 44(1):1–36, 2015.] that characterises the finite-domain languages that are correctly solved (in polynomial time) by the BLP. Their characterisation is given in terms of algebraic properties of the valued constraint language. We will give sufficient conditions for an infinite-domain valued constraint language $\Gamma$ to be correctly solved (in polynomial time) by applying the BLP to a suitable substructure of $\Gamma$.

This is a joint work with Manuel Bodirsky and Marcello Mamino.

## Similarity, critical relations, and Zhuk's bridges

One of the most celebrated events in universal algebra in recent memory was the simultaneous announcement in 2017 of two independent positive solutions to the Constraint Satisfaction Dichotomy Conjecture by Andrei Bulatov and Dmitriy Zhuk. A key aspect of each proof is an analysis of implicit linear relations in CSP instances compatible with a Taylor algebra. In this lecture I will explore Zhuk's version of this analysis and argue that it can be viewed as an extension of the similarity relation (from commutator theory for congruence modular varieties) to locally finite Taylor varieties.

## Transitive systems

By a transitive system we understand a tuple $(X, \mathcal{R}, f, G)$ where $X$ is a nonempty set, $\mathcal{R}$ is a reflexive and transitive relation on $X$, $G$ is an abelian group and $f\colon \mathcal{R} \rightarrow G$ is a transitive function (i.e. for every $x\in X$, $f(x,x)=0$ and if $(x,y), (y,z)\in \mathcal{R}$ then $f(x,y)+f(y,z)=f(x,z)$). This concept emerged naturally in determining the isomorphisms between full algebras of matrices and is conveniently describable by means of the graph theory. An important and intriguing issue is a possibility of completion of a given transitive system. We say that the system can be completed if the function $f$ can be extended to a transitive function $\hat{f}\colon X\times X\rightarrow G$. Of particular interest are relations $\mathcal{R}$ that predetermine possibility of completion of the transitive system independently on the nontrivial abelian group $G$. We present several classes and examples of such relations together with the general methods of this theory.

## Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

We study the amount of consistency (measured by relational width) needed to solve CSPs for first-order reducts of homogeneous graphs with bounded strict width that always contain the edge relation $E$, the non-edge relation $N$ and the equality.

It is known that every countably infinite homogeneous graph $G$ is finitely bounded, that is, there exists a finite set $\operatorname{BOUNDS}(G)$ such that a finite graph $G'$
embeds into $G$ if and only if there exists no $H$ in $\operatorname{BOUNDS}(G)$ such that $H$ embeds into $G$.

Our main result is that the structures under consideration have relational width exactly $(2, M)$ where $M$ is the biggest size of a graph in $\operatorname{BOUNDS}(G)$.

## On the enhanced power graph of a finite group

The directed power graph of a group was introduced by Kelarev and Quinn as the simple digraph whose vertex set is the universe of the group, and with $x\rightarrow y$ if $y$ is a power of $x$, while the power graph of a group as the underlying undirected graph of the directed power graph. The enhanced power graph of a group, which was introduced by Cameron et al., is the graph with the same vertex set such that two vertices $x$ and $y$ are adjacent if they are both powers of some element $z\in G$.

Cameron proved that the power graph of a finite group determines the directed power graph. We prove that the enhanced power graph determines the directed power graph too. Beside that, we describe the enhanced power graphs of finite abelian groups, determine all finite groups whose enhanced power graph has automorphism group with some properties, and we investigate some combinatorial properties of said graph.

## The complexity of the Quantified Constraint Satisfaction Problem on a 3-element set

The Quantified Constraint Satisfaction Problem $\operatorname{QCSP}(\Gamma)$ is the problem to evaluate a sentence of the form $\forall x_1 \exists y_1 \dots \forall x_n \exists y_n \ (R_{1}(\dots)\wedge\dots\wedge R_{s}(\dots))$, where $R_1,\dots,R_s$ are relations from the constraint language $\Gamma$. This problem is a generalization of the Constraint Satisfaction Problem, where only existential quantifiers are allowed.

We classified the complexity of the Quantified Constraint Satisfaction Problem for constraint languages on 3-element domain containing all unary singleton relations (so called idempotent case), i.e. we showed that for such languages $QCSP(\Gamma)$ is either tractable, or NP-complete, or coNP-complete, or PSpace-complete.

Moreover, we showed that there exists a constraint language $\Gamma$ on 4-element domain such that the problem $\operatorname{QCSP}(\Gamma)$ is DP-complete (from Boolean Hierarchy), and a constraint language $\Gamma$ on 13-element domain giving another complexity class different from all the above classes.

## A Code Equivalence Theorem for Infinite Rings

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams theorem on code equivalence, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.

This is joint work with Friedrich Martin Schneider.