# Csaba SzaboEötvös Loránd University · Institute of Mathematics

Csaba Szabo

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66

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Introduction

**Skills and Expertise**

## Publications

Publications (66)

For an algebra [Formula: see text] the algebra [Formula: see text] is called a functional reduct if each [Formula: see text] is a term function of [Formula: see text]. We classify the functional reducts of the countable atomless Boolean algebra up to first-order interdefinability. That is, we consider two functional reducts the “same” if their grou...

Retrieving information from memory can-under many circumstances-strengthen one's memory of the retrieved information itself. The strategic use of retrieval to enhance memory and help long-term retention is known as retrieval practice. However , it is unclear whether its effect also holds true in the case of learning mathematics. This research is an...

We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These groups correspond, up to first-order interdefinability, precisely to all structures with a first-order defini...

The impact playing has on the development of thinking is an important topic of psychology of learning, brain research and mathematics didactics.Our research is also connected to the aforementioned topic. We investigated the effects of playing board games on competence motivation and the development of mathematical competencies.In this paper, we pre...

A játékosítást több mint tíz éve használják a vállalati kultúrában, ezzel növelik a produktivitást, az alkalmazottak motivációját és elköteleződését. Hasonló hatás érhető el a tanulási folyamatokban is. A nevelésben a játékosítással elért elkötelezettség növeli a diákok teljesítményét és javítja a tanuláshoz való hozzáállásukat. A játékosítás hatás...

Mathematik ist in Ungarn traditionell von hoher kultureller und wissenschaftlicher Bedeutung. Intention der Buchreihe „Mathematiklehren und -lernen in Ungarn“ ist es, die beispielgebende Rolle des Landes und den inspirativen Austausch über Grenzen hinweg zum Ausdruck zu bringen. Der vorliegende Band enthält – ganz in diesem Sinne – Artikel aus mehr...

Studying international literature on the psychology of learning, brain research, and didactic we conclude that logical thinking can be improved through playing board games. The board game player performs complex logical operations and can enjoy the game with little factual knowledge. (Herber et al. 2003) Our research is based on a school experiment...

A felgyorsult technikai fejlődés és a munkaerő piaci elvárások a felsőoktatást új kihívások elé állítják. A kooperatív munka jelen van a termelés minden szintjén. Mindezekre egyfajta válasz lehet a kooperatív tanulás, mint oktatási módszer bevezetése a felsőoktatásban. A kooperatív tanulás megfelelő előkészítéssel hatékonyabb a felsőoktatásban jele...

The curricular background of the transition problem from highschool to universty is analysed in Hungary. While students finish their mathematical studies successfully at highschool, pass their final exams, this knowledge seems to disappear at their first year at university. We investigate the mathematical knowledge expected by the Hungarian univers...

The description of the poset of clones generated by a single binary idempotent monomial over \(\mathbb {F}_q\) is given by purely number theoretic means.

In this paper we present recursive formulas to compute the fine spectrum and generative spectrum of each of the varieties of monounary algebras. Hence, an asymptotic or log-asymptotic estimation for the number of n-generated and n-element algebras is given in every variety of monounary algebras. These results provide infinitely many examples of spe...

We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics, which may be of independent interest.

It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts.

In this paper we classify some special reducts of the countable atomless
Boolean algebra which we call functional reducts. We prove that there are
exactly $13$ such structures up to first order interdefinability.

Let $\mathbb{F}_2^\omega$ denote the countably infinite dimensional vector
space over the two element field and $\operatorname{GL}(\omega, 2)$ its
automorphism group. Moreover, let $\operatorname{Sym}(\mathbb{F}_2^\omega)$
denote the symmetric group acting on the elements of $\mathbb{F}_2^\omega$. It
is shown that there are exactly four closed subg...

In this paper it is shown that the logarithm of the number of non-isomorphic rooted trees of depth $k\geq 3$ is asymptotically $\frac{\pi^2}{6}\frac{n}{\log \log\cdots \log n}$, where $\log$ is iterated $k-2$ times in the denominator.

It is observed in this paper that the complexities of the equivalence and the equation solvability problems are not determined by the clone of the algebra. In particular, we prove that for the alternating group on four elements these problems have complexity in P; if we extend the group by the commutator as an extra operation, then the equivalence...

Recently it has been shown that all non-trivial closed permutation groups
containing the automorphism group of the random poset are generated by two
types of permutations: the first type are permutations turning the order upside
down, and the second type are permutations induced by so-called rotations. In
this paper we introduce rotations for finit...

The aim of the present paper is to carry on the research of Czédli in determining the maximum number of rectangular islands on a rectangular grid. We estimate the maximum of the number of square islands on a rectangular grid.

Piecewise testable languages are widely studied area in the theory of automata. We analyze the algebraic properties of these languages via their syntactic monoids. In this paper a normal form is presented for 2- and 3-piecewise testable languages and a log-asymptotic estimate is given for the number of words over these monoids.

We determine, up to the equivalence of first-order interdefinability, all
structures which are first-order definable in the random partial order. It
turns out that these structures fall into precisely five equivalence classes.
We achieve this result by showing that there exist exactly five closed
permutation groups which contain the automorphism gr...

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity
International audience
We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time...

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity
International audience
For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunt...

In this note we give an asymptotic estimate for the number of finite non-isomorphic monounary algebras of a given size.

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations onA. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that
every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p
n
-sequence...

We investigate the computational complexity of deciding whether or not a given polynomial, presented as the sum of monomials, is identically 0 over a ring. It is proved that if the factor by the Jacobson-radical is not commutative, then the problem is coNP-complete.

We present a new solution of the word problem of free algebras in varieties generated by iterated semidirect products of semilattices. As a consequence, we provide asymptotical bounds for free spectra of these varieties. In particular, each finite R-trivial (and, dually, each finite L-trivial) semigroup has a free spectrum whose logarithm is bounde...

The aim of the present paper is to carry on the re- search of Czédli in determining the maximum number of rectan- gular islands on a rectangular grid. We estimate the maximum of the number of square islands on a rectangular grid.

Let a height function f be a real valued function on R2. A connected subset of R2 is called an island, if there is a water level such that H is an island in the classical sense. We show that an island system is always laminar. Among others, in this paper we prove that the cardinality of a maximal laminar system is either countable or continuum.

In this short note we give an asymptotic formula for the p
n
sequence of the variety of bands, namely,
pn(B)=\frac1n2K2n+1(1+O(\frac1n)),p_{n}(\mathcal{B})=\frac{1}{n^{2}}K^{2^{n+1}}(1+O(\frac{1}{n})),
for some constant K. This yields a formula for the free spectrum of this variety.

We analyze the free spectra of semigroup varieties generated by finite combinatorial 0-simple semigroups. We give estimates
for all of them.

The equivalence problem for a group G is the problem of deciding which equations hold in G. It is known that for finite nilpotent groups and certain other solvable groups, the equivalence problem has polynomial-time
complexity. We prove that the equivalence problem for a finite nonsolvable group G is co-NP-complete by reducing the k-coloring proble...

We give an asymptotic bound for the size of the n-generated relatively free semigroup in the variety generated by all combinatorial strictly 0-simple semigroups.(Received March 10 2006)(Revised September 16 2006)(Accepted March 15 2006)

We analyze the computational complexity of solving a single equation and checking identities over finite meta-abelian groups. Among others we answer a question of Goldmann and Russel from 1998: we prove that it is decidable in polynomial time whether or not an equation over the six-element group S3 has a solution.

We give an asymptotical bound for the logarithm of the free spectrum of the variety generated by the five element Brandt semigroup.
Our bound is much better than any previously known estimate.

In this article we investigate the structure of rings with some strong symmetry condition.

In this paper we analyze the so-called word problem for (finite)
combinatorial 0-simple semigroups and matrix semigroups from the viewpoint
of computational complexity.

Chomp is a 50 year-old game played on a partially ordered set P. It has been in the center of interest of several mathematicians since then. Even when P is simply a 3 ! n lattice, we have almost no information about the winning strategy. In this paper we present a new approach and a cubic algorithm for computing the winning positions for this case....

We analyze the so-called word-problem for M2(Z2), the ring of 2 × 2 matrices over Z2. We prove that the term-equivalence problem for the semigroup (and so for the ring) M 2(Z2) is coNP-complete.

We complete the investigations into the word-problem for finite matrix rings.
Namely we prove that M2
(Z3), the ring of 2 x 2
matrices over Z3, has a
coNP-complete term-equivalence (or identity checking) problem.

We analyze the so-called word-problem for M2(Z2), the ring of 2 2m atrices overZ2. We prove that the term-equivalence problem for the semigroup (and so for the ring) M2(Z2 )i s coNP-complete. In this paper we study the computational complexity of the word-problem for M2(Z2). We shall use the standard notation for computational complexity, as P, NP,...

We construct monotone Jónsson terms and near-unanimity functions based on combinatorial properties of finite posets.

In this paper, we connect the constraint satisfaction problem with other complexity problems, like the polynomial equivalence problem for combinatorial 0-simple semigroups, the graph retraction problem and the geometry problem. We show that every constraint satisfaction problem is polynomially equivalent to an easily formulated algebra complexity p...

Davey and Quackenbush proved a strong duality for each dihedral group Dm with m odd. In this paper we extend this to a strong duality for each finite group with cyclic Sylow subgroups (such groups are known to be metacyclic).

It is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is du...

For a graph G, OALG asks whether or not an input graph H together with a partial map g:S→G, S⊆V(H), admits a homomorphism f:H→G such that f|S=g. We show that for connected graphs G1, G2, OAL G1×G2 is in P if G1 and G2 are trees and NP-complete otherwise.

Let R be a finite commutative ring with identity. If the Jacobson radical of R annihilates itself, then the quasivariety generated by R is dually equivalent to a category of structured Boolean spaces obtained in a natural way from R. If on the other hand the radical of R does not annihilate itself, then no such natural dual equivalence is possible....

An n-ary operation f is totally symmetric if it obeys the identity f(x
1,...,x
n
)=f(y
1,...,y
n
) for all sets of variables such that {x
1,...,x
n
}={y
1,...,y
n
}. We characterize finite posets admitting an n-ary idempotent totally symmetric operation for all n. The characterization is expressed in terms of zigzags, special objects related to...

In this paper we introduce a new version of the concept of order varieties. Namely, in addition to closure under retracts and products we require that the class of posets should be closed under taking idempotent subalgebras. As an application we prove that the variety generated by an order-primal algebra on a finite connected poset P is congruence...

In this paper we show that a finite nilpotent ring that is not a zero-ring cannot admit a natural duality. In fact, every
finite ring having a nilpotent subring (which is nilpotent of class ≥ 2) is not dualizable.

An independence algebra is an algebra A in which the subalgebras satisfy the exchange axiom, and any map from a basis of A into A extends to an endomorphism. Independence algebras fall into two classes; the first are specified by a set X , a group G, and a G-space C. The second are much more restricted; we show that the subalgebra lattice is a proj...

Complexity problems associated with finite rings and finite semigroups, particularly semigroups of matrices over a field and the Rees matrix semigroups, are examined. Let M_nF be the ring of n x n matrices over the finite field F and let T_nF be the multiplicative semigroup of n x n matrices over the finite field F. It is proved that for any finite...

In this paper we show that Z_8 does not admit a natural duality. In fact, we show that 2Z_8 = {2, 4, 6, 8 | +,.} is not dualizable, and this will imply that the original ring is not dualizable, either. As a corollary we show that Sindi's conjecture does not hold. Our technique will be similar to one due to Quackenbush and Szab\'o, where non-dualiza...

Változatos kérdéseket vizsgáltunk a csoportelméletben, a csoportok reprezentációelméletében és más kapcsolódó absztrakt algebrai területeken. 39 tudományos dolgozatot publikáltunk, ezek nagy részét vezető nemzetközi folyóiratokban (pl. Bulletin of the London Mathematical Society, Duke Mathematical Journal, European Journal of Combinatorics, Journal...

Keith Kearnes és Kiss Emil könyvükben a kongruenciaszelídítés néhány eredményét terjesztik ki nem lokálisan véges varietásokra. Jellemzik azokat a varietásokat, melyekben teljesül nemtriviális kongruencia-azonosság, a négyszögletes kommutátorral, tiltott részhálóval és Malcev-feltételekkel is. Belátják, hogy ha egy ilyen varietás reziduálisan kicsi...