
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Abstract. Some elements of tame congruence theory can be applied to quasiorder lattices instead of congruence lattices. In particular, it is possible to consider minimal sets of an algebra with respect to one of its prime quasiorder quotients. It turns out that if a finite algebra is in a congruence modular variety, then it is minimal with respect to a quasiorder quotient iff it is minimal with respect to a congruence quotient  in which case it is either a twoelement algebra, or has a Mal'tsevpolynomial. As an application of this fact, we prove that if an algebra is in a congruence modular variety, its congruence and quasiorder lattices satisfy the same identities.
DOI: 10.14232/actasm0180244
AMS Subject Classification
(1991): 08A30, 08B10, 06B15
Keyword(s):
quasiorder,
modularity,
lattice identity
received 22.2.2018, revised 27.1.2020, revised 9.3.2020, accepted 11.3.2020. (Registered under 24/2018.)
Abstract. Nearrings are the nonlinear generalization of rings. Planar nearrings play an important role in nearring theory, both from the structural side, being close to generalized nearfields, as well as from an applications perspective, in geometry and combinatorial designs related to difference families. In this paper we investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of an abelian group by its zero multiplier part. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring, a coordinatisation result. This generalizes the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized centre of planar nearrings as well as finding new proofs of older results.
DOI: 10.14232/actasm018036y
AMS Subject Classification
(1991): 16Y30
Keyword(s):
distributivity,
planar nearring,
planar ring,
nearfield,
nearvector space,
generalized centre
received 9.4.2018, revised 8.6.2019, accepted 18.11.2019. (Registered under 36/2018.)
Erkko Lehtonen,
Reinhard Pöschel

3150

Abstract. Introduced by C.~R. Shallon in 1979, graph algebras establish a useful connection between graph theory and universal algebra. This makes it possible to investigate graph varieties and graph quasivarieties, i.e., classes of graphs described by identities or quasiidentities. In this paper, graph quasivarieties are characterized as classes of graphs closed under directed unions of isomorphic copies of finite strong pointed subproducts.
DOI: 10.14232/actasm0195289
AMS Subject Classification
(1991): 05C25, 08C15
Keyword(s):
graph algebras,
quasivarieties
received 29.5.2019, revised 4.3.2020, accepted 6.3.2020. (Registered under 528/2019.)
Orest D. Artemovych,
Victor A. Bovdi,
Mohamed A. Salim

5172

Abstract. Let $R[G]$ be the group ring of a group $G$ over an associative ring $R$ with unity such that all prime divisors of orders of elements of $G$ are invertible in $R$. If $R$ is finite and $G$ is a Chernikov (torsion $FC$) group, then each $R$derivation of $R[G]$ is inner. Similar results also are obtained for other classes of groups $G$ and rings $R$.
DOI: 10.14232/actasm019664x
AMS Subject Classification
(1991): 20C05, 16S34, 20F45, 20F19, 16W25
Keyword(s):
group ring,
derivation,
locally finite group,
solder,
torsionfree group,
nilpotent group,
differentially trivial ring,
nilpotent Lie ring,
solvable Lie ring
received 18.6.2019, revised 19.2.2020, accepted 20.2.2020. (Registered under 664/2019.)
Gábor Bacsó,
Zsolt Tuza

7379

Abstract. Given a set $W$ of positive integers, a set $I\subseteq W$ is \emph {independent} if all the partial sums in $I$ are distinct. We prove estimates on the maximum size of an independent set within a set of $n$ integers.
DOI: 10.14232/actasm0198715
AMS Subject Classification
(1991): 11B75, 05D99
Keyword(s):
equal subset sums,
sumindependence of sets
received 21.6.2019, accepted 9.10.2019. (Registered under 621/2019.)
Kirby A. Baker,
George Grätzer

81104

Abstract. Zilber's Theorem states that a finite lattice $L$ is planar if and only if it has a complementary order relation. We provide a new proof for this crucial result and discuss some applications, including a canonical form for finite planar lattices and an analysis of coverings in the leftright order.
DOI: 10.14232/actasm0192309
AMS Subject Classification
(1991): 06C10, 06A07
Keyword(s):
planar lattice,
Zilber's Theorem
received 30.7.2019, revised 22.1.2020, accepted 29.1.2020. (Registered under 730/2019.)
Christian Herrmann,
Niklas Niemann

105115

Abstract. We show that a subdirectly irreducible $*$regular ring admits a representation within some inner product space provided so does its ortholattice of projections.
DOI: 10.14232/actasm019837x
AMS Subject Classification
(1991): 06C20; 16E50, 16W10
Keyword(s):
orthocomplemented modular lattice,
$*$regular ring,
frame,
inner prodcut space,
representation
received 7.8.2019, revised 19.1.2020, accepted 12.2.2020. (Registered under 87/2019.)
Abstract. Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite joinenriched meet semilattices, and chopped lattices. We prove that if an $n$element nearlattice has at least $83\cdot 2^{n8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp.
DOI: 10.14232/actasm0195734
AMS Subject Classification
(1991): 06A12, 06B75, 20M10
Keyword(s):
planar nearlattice,
planar semilattice,
planar lattice,
chopped lattice,
number of subalgebras,
computerassisted proof,
commutative idempotent semigroup
received 23.8.2019, revised 22.1.2020, accepted 29.1.2020. (Registered under 823/2019.)
Takeshi Yoshimoto

167182

Abstract. We prove a multiparameter $L\log ^{k}L$ generalization of the BáezDuarte Abelian ergodic theorem for positive linear contractions on $L_{1}$, which allows the application of the local convergence principle of Sucheston's type. Next we establish a new weighted Abelian ratio ergodic theorem for DunfordSchwartz operators on $L_{1}$ with modulation by Besicovitch sequences. Moreover, this (oneparameter) result is generalized to the case of multiparameter operator averages, which allows the application of Fava's maximal ergodic inequality.
DOI: 10.14232/actasm0197574
AMS Subject Classification
(1991): 47A35, 40H05, 40G10
Keyword(s):
local convergence principle,
Chacon's theorem,
BáezDuarte's theorem,
Fava's theorem,
Abelian ratio ergodic theorem,
positive linear contraction,
Besicovitch sequence,
modulated Abelian ratio ergodic theorem
received 21.1.2019, revised 25.11.2019, accepted 4.3.2020. (Registered under 7/2019.)
Abstract. We shall introduce the notion of the Picard group for an inclusion of $C^*$algebras. We shall also study its basic properties and the relation between the Picard group for an inclusion of $C^*$algebras and the ordinary Picard group. Furthermore, we shall give some examples of the Picard groups for unital inclusions of unital $C^*$algebras.
DOI: 10.14232/actasm0192711
AMS Subject Classification
(1991): 46L05
Keyword(s):
equivalence bimodules,
inclusions of $C^*$algebras,
the Picard group
received 22.3.2019, revised 16.4.2019, accepted 13.5.2019. (Registered under 21/2019.)
Abstract. We study the weighted composition operators between the Lipschitz space and the space of bounded functions on the set of vertices of an infinite tree. We characterize the boundedness, the compactness, and the boundedness from below of weighted composition operators. We also determine the isometric weighted composition operators.
DOI: 10.14232/actasm0195226
AMS Subject Classification
(1991): 47B33, 47B38, 05C05
Keyword(s):
tree,
Lipschitz space,
weighted composition operator
received 24.3.2019, revised 11.3.2020, accepted 20.3.2020. (Registered under 22/2019.)
Abstract. Let $\Gamma ^n_k$ be the space of all the $k$dimensional totally geodesic submanifolds of the $n$dimensional real hyperbolic space where $1\leq k\leq n1$. We prove that the Radon transform $R$ for double fibrations of the real hyperbolic Grassmann manifolds $\Gamma ^n_p$ and $\Gamma ^n_q$ with respect to the inclusion incidence relations maps $C^\infty _0(\Gamma ^n_p)$ bijectively onto the space of all the functions in $C^\infty _0(\Gamma ^n_q)$ which satisfy a certain system of linear partial differential equations explicitly constructed from the left infinitesimal action of the transformation group when $0\leq p<q\leq n1$ and $\dim \Gamma ^n_p< \dim \Gamma ^n_q$. Our approach is based on the generalized method of gnomonic projections. We also treat the dual Radon transform $R^*$.
DOI: 10.14232/actasm0197731
AMS Subject Classification
(1991): 44A12; 43A85
Keyword(s):
Radon transform,
real hyperbolic space,
Grassmann manifold
received 26.3.2019, revised 2.9.2019, revised 17.3.2020, accepted 18.3.2020. (Registered under 23/2019.)
S. Kar,
P. Veeramani

265271

Abstract. We prove a best proximity point version of Krasnoselskii's fixed point theorem. As a consequence we obtain the existence of best proximity points of relatively \emph {ucontinuous} maps for a pair of compact convex sets.
DOI: 10.14232/actasm0190184
AMS Subject Classification
(1991): 47H09, 47H10; 46B20
Keyword(s):
best proximity point,
metric projection,
property UC
received 18.5.2019, revised 2.12.2019, accepted 10.12.2019. (Registered under 518/2019.)
N. Mallick,
K. Sumesh

273286

Abstract. We introduce the notion of $Q$commuting operators which includes commuting operators. We prove a generalized version of the commutant lifting theorem and Ando's dilation theorem in the context of $Q$commuting operators.
DOI: 10.14232/actasm0197752
AMS Subject Classification
(1991): 47A20
Keyword(s):
isometric dilation,
unitary dilation,
coisometric extension,
commutant lifting,
Ando's dilation theorem
received 25.5.2019, revised 21.10.2019, accepted 28.10.2019. (Registered under 525/2019.)
Abstract. In this paper we prove that there exists a continuous function on $[0,1)^2$, with a certain smoothness, whose double FourierWalshPaley series diverges by rectangles on a set of positive measure.
DOI: 10.14232/actasm0193190
AMS Subject Classification
(1991): 42C10
Keyword(s):
WalshPaley,
double Fourier series,
divergence a.e
received 9.6.2019, revised 8.1.2020, accepted 14.1.2020. (Registered under 69/2019.)
Abstract. The concepts of multiresolution analysis (MRA) and wavelet have been generalized to a local field $K$ of positive characteristic by using a prime element $\mathfrak p$ of such a field. A MRA is a sequence of closed subspaces of $L^2(K)$ satisfying certain properties. In this paper, we are interested in a nonstationary MRA and related wavelets on local fields of positive characteristic.
DOI: 10.14232/actasm0191189
AMS Subject Classification
(1991): 42C40, 42C15, 43A70, 11S85
Keyword(s):
wavelet,
nonstationary,
multiresolution analysis,
local field,
Fourier transform
received 18.6.2019, revised 30.7.2019, accepted 31.7.2019. (Registered under 618/2019.)
Horst Martini,
Zokhrab Mustafaev

321330

Abstract. Some sharp bounds for the inner radius and the outer radius of the unit ball of a (normed or) Minkowski space with respect to its isoperimetrix are known. To find more such bounds is a challenging problem. Related to this motivation, we derive new sharp inequalities between inner and outer radii for the HolmesThompson and Busemann measures. Crosssection measures as well as the BlaschkeSantaló inequality will be used to obtain these new inequalities.
DOI: 10.14232/actasm0191304
AMS Subject Classification
(1991): 46B20, 52A20, 52A21, 52A40
Keyword(s):
affine isoperimetric inequalities,
BlaschkeSantalo inequality,
Busemann measure,
crosssection measures,
HolmesThompson measure,
inner radius,
intersection body,
isoperimetrix,
Minkowski geometry,
outer radius,
projection body
received 30.6.2019, revised 23.9.2019, accepted 25.9.2019. (Registered under 630/2019.)
Rupert Lasser,
Josef Obermaier

331342

Abstract. The main purpose of this paper is to use chain sequences to study spectral properties of weighted shift operators $A$ and of tridiagonal operators $\Re A$. Characterizations of chain sequences and relations to Haar sequences are derived. We use these results to compare the spectral radius, the numerical radius and the norm of $A$ and $\Re A$. As an example we study orthogonal polynomials defined by a recursion formula with almost constant coefficients.
DOI: 10.14232/actasm0191524
AMS Subject Classification
(1991): 47B36, 47B37, 33C45
Keyword(s):
weighted shift operators,
Jacobi operators,
chain sequences,
orthogonal polynomials
received 21.11.2019, accepted 20.2.2020. (Registered under 152/2019.)
