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 two-element algebra, or has a Mal'tsev-polynomial. 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/actasm-018-024-4

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/actasm-018-036-y

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.)

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 quasi-identities. 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/actasm-019-528-9

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.)

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/actasm-019-664-x

AMS Subject Classification (1991): 20C05, 16S34, 20F45, 20F19, 16W25

Keyword(s): group ring, derivation, locally finite group, solder, torsion-free 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.)

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/actasm-019-871-5

AMS Subject Classification (1991): 11B75, 05D99

Keyword(s): equal subset sums, sum-independence of sets

received 21.6.2019, accepted 9.10.2019. (Registered under 621/2019.)

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 left-right order.

DOI: 10.14232/actasm-019-230-9

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.)

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/actasm-019-837-x

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 join-enriched meet semilattices, and chopped lattices. We prove that if an $n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp.

DOI: 10.14232/actasm-019-573-4

AMS Subject Classification (1991): 06A12, 06B75, 20M10

Keyword(s): planar nearlattice, planar semilattice, planar lattice, chopped lattice, number of subalgebras, computer-assisted proof, commutative idempotent semigroup

received 23.8.2019, revised 22.1.2020, accepted 29.1.2020. (Registered under 823/2019.)

Abstract. We prove a multiparameter $L\log ^{k}L$ generalization of the Báez--Duarte 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 Dunford--Schwartz operators on $L_{1}$ with modulation by Besicovitch sequences. Moreover, this (one-parameter) result is generalized to the case of multiparameter operator averages, which allows the application of Fava's maximal ergodic inequality.

DOI: 10.14232/actasm-019-757-4

AMS Subject Classification (1991): 47A35, 40H05, 40G10

Keyword(s): local convergence principle, Chacon's theorem, Báez-Duarte'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/actasm-019-271-1

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/actasm-019-522-6

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 $1\leq k\leq n-1$. 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 $0\leq p<q\leq n-1$ 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/actasm-019-773-1

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.)

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 {u-continuous} maps for a pair of compact convex sets.

DOI: 10.14232/actasm-019-018-4

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.)

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/actasm-019-775-2

AMS Subject Classification (1991): 47A20

Keyword(s): isometric dilation, unitary dilation, co-isometric 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 Fourier--Walsh--Paley series diverges by rectangles on a set of positive measure.

DOI: 10.14232/actasm-019-319-0

AMS Subject Classification (1991): 42C10

Keyword(s): Walsh--Paley, 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/actasm-019-118-9

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.)

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 Holmes--Thompson and Busemann measures. Cross-section measures as well as the Blaschke--Santaló inequality will be used to obtain these new inequalities.

DOI: 10.14232/actasm-019-130-4

AMS Subject Classification (1991): 46B20, 52A20, 52A21, 52A40

Keyword(s): affine isoperimetric inequalities, Blaschke--Santalo inequality, Busemann measure, cross-section measures, Holmes--Thompson 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.)

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/actasm-019-152-4

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.)