# Scientific Journals and Yearbooks Published at SAS

## Mathematica Slovaca

Volume 52, 2002, No. 2

Content:

Radicals in non-commutative generalizations of $MV$@-algebras
Jiří Rachůnek 1)

 1) Katedra algebry a geometrie, PřF UP; Tomkova 40; 779 00 Olomouc; ČR.

$GMV$@-algebras are a non-commutative generalization of $MV$@-al ge bras. In the paper, structure properties of $GMV$@-algebras are studied using a duality between $GMV$@-algebras and unital $\ell$@-groups. Three kinds of radicals of $GMV$@-algebras are investigated in general and, particularly, in the cases of finite valued and normal-valued $GMV$@-algebras. A class of $GMV$@-algebras possessing infinitely many state-morphisms is described.

Mathematica Slovaca. Volume 52, 2002, No. 2: 135-144.

Zero invariant and idempotent fuzzy BCC-subalgebras
Wiesław A. Dudek 1), Young Bae Jun 2)

 1) Technical University of Wrocław, Institute of Mathematics; W. Wyspiańskiego 27; 50-370 Wrocław; Poland. 2) Dept. of Math. Education, Gyeongsang National University; Chinju 660-701; Korea.

We introduce the notion of zero invariant and idempotent fuzzy BCC@-subalgebras in BCC@-algebras with respect to an $s$-norm, and investigate some of their properties.

Mathematica Slovaca. Volume 52, 2002, No. 2: 145-156.

Algebraic properties of pre-logics
Ivan Chajda 1), Radomír Halaš 2)

 1) Katedra algebry a geometrie, UP Olomouc; Tomkova 40; 779 00 Olomouc; ČR. 2) Katedra algebry a geometrie, Přírodovědecká fakulta;Univerzita Palackého Olomouc; Tomkova 40; CZ–779 00 Olomouc; Česká Republika.

We introduce the concept of a pre-logic which is an algebra weaker than a Hilbert algebra (an algebraic counterpart of intuitionistic logic) but strong enough to have deductive systems. On every such a pre-logic $A$ a quasiorder $Q$ can be defined and a Hilbert algebra can be reached as a quotient algebra of $A$ by the congruence induced by $Q$. We study algebraic properties of pre-logics and of lattices of their deductive systems.

Mathematica Slovaca. Volume 52, 2002, No. 2: 157-175.

A note on two circumference generalizations of Chvátal's hamiltonicity condition

 1) Odd. informatiky MÚ SAV, Dúbravská cesta 9; 841 05 Bratislava.

In his book [BOLLOBÁS, B.: Extremal graph theory, Academic Press, London-New York-San Francisco, 1978] the author asks about possible generalizations of Chvátal's well-known hamiltonicity condition [CHVÁTAL, V.: On hamilton's ideals , J. Combin. Theory Ser. B 12 (1972), 163–168]. For $c=3$ and $4$ this follows directly from $2$@-connectivity. However, Häggkvist [Personal communication with J. A. Bondy] found counterexamples for any $c≥ 7$. In this paper we treat the remaining cases and show that for $c=5$ such generalization is possible while for $c=6$ we give counterexamples. Moreover, we show that some circumference generalization of Chvátal's condition for any $c$ is even possible.

Mathematica Slovaca. Volume 52, 2002, No. 2: 163-168.

Two element direct limit classes of monounary algebras
Emília Halušková 1)

 1) Matematický ústav SAV, Grešákova 6; 040 01 Košice.

A class of algebras is said to be direct limit closed if it is closed with respect to direct limits. We describe all two element sets $S$ of monounary algebras such that $S$, together with all isomorphic copies of elements of $S$, is a direct limit closed class.

Mathematica Slovaca. Volume 52, 2002, No. 2: 177-194.

Metric distribution results for sequences $(\{qn\vec{α}\})$
Hansjörg Albrecher 1)

 1) Department of Mathematics, Graz University of Technology; Steyrergasse 30; A–8010 Graz; AUSTRIA.

In this paper a recent result of Philipp and Tichy (2000) on the well-distribution measure of certain binary pseudorandom sequences in the unit interval is generalized. Furthermore the average value of the $L2$@-discrepancy of sequences $(\{qn\vec{α}\})n≥ 1$ is calculated, where $(qn)n≥ 1$ is a given sequence of positive integers and $\vec{α}\in[0,1]d$.

Mathematica Slovaca. Volume 52, 2002, No. 2: 195-206.

On asymptotic behavior of solutions of some difference equation
Andrzej Drozdowicz 1), Janusz Migda 2)

 1) Poznań University of Technology, Institute of Mathematics; ul. Piotrowo 3a; 60-965 Poznań; Poland. 2) Faculty of Mathematics and Computer Sciences, A.Mickiewicz University; Poznań; POLAND.

Asymptotic properties of solutions of difference equations of the form

$$Δ2xn =anφ(xn+p+1,…,xn+p+k)+bn$$

are studied.

Mathematica Slovaca. Volume 52, 2002, No. 2: 207-214.

Once more on the double Fourier-Haar series
Ondrej Kováčik 1)

 1) Katedra matematiky SvF, Žilinská univerzita; J. M. Hurbana 15; 010 26 Žilina.

This paper deals with a necessary and sufficient condition for integrability of the majorants $P\kern -1pt f$ of the orthogonal partial sums of Fourier-Haar series.

Mathematica Slovaca. Volume 52, 2002, No. 2: 215-220.

On the Lefschetz fixed point theorem
Lech Górniewicz 1)

 1) Schauder Center for Nonlinear Studies, Nicholas Copernicus University; Chopina 12/18; PL–87-100 Toruń; POLAND.

The aim of this paper is to present current results concerning the Lefschetz Fixed Point Theorem for metric spaces. Some new results are included. In particular, an abstract version and also the Lefschetz Fixed Point Theorem for condensing mappings are proved.

Mathematica Slovaca. Volume 52, 2002, No. 2: 221-233.

An occupancy problem with group drawings of different sizes
Saleh M. Mitwalli 1)

 1) Dept. of Math., Faculty of Science; P.O.: 34517(3),; New Damiatta; Egypt.

We consider the classical occupancy problem with group drawings of different sizes. We derive the exact distribution of the number of balls occurring in all sampled groups and its asymptotic behavior. Furthermore, the number of drawings until at most $k$ balls are observed in all samples is studied.

Mathematica Slovaca. Volume 52, 2002, No. 2: 235-242.