Volume 68, Issue 6

Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen , Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups , Czechoslovak Math. J. 3 (1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.

It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

J. Jakubík noted in [JAKUBÍK, J.: Modular Lattice of Locally Finite Length , Acta Sci. Math. 37 (1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic to S 7 . In the present paper we extend the scope of Šik’s theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.: Modular Lattice of Locally Finite Length , Acta Sci. Math. 37 (1975), 79–82] cannot be strengthened is analogous way.

In this paper we define and study the weak pseudo-BCK algebras as generalizations of weak BCK-algebras, extending some results given by Cı⃖rulis for weak BCK-algebras. We give some characterizations of weak pseudo-BCK algebras and we prove that a weak pseudo-BCK algebra satisfying the right distribution laws is a BCK-algebra. We define the class of commutative weak pseudo-BCK algebras, and we give equivalent definitions and characterization theorems for commutative weak pseudo-BCK algebras. The classes of quasi pseudo-BCK algebras and weak pseudo-BCK(E) algebras are introduced and a characterization theorem for quasi pseudo-BCK algebras is given. We prove that any weak pseudo-BCK(E) algebra is a pseudo-BE algebra and the class of commutative weak pseudo-BCK(E) algebras is equivalent to the class of commutative pseudo-BCK algebras.

The main theorem of the paper gives an example of a non-maximal order 𝓞 in a quadratic number field K such that the homomorphism W 𝓞 → WK of Witt rings is injective.

An old question, due to Diophantus, asks to find sets of rational numbers such that 1 added to the product of any two elements from the set is a square. We are concerned here with a modification of this question. Let t ≥ 2 be an integer, and let 𝔽 be a field. For d ∈ 𝔽, define f t , d : 𝔽 t → 𝔽 as ft,d(x1,x2,…,xt):=x1x2⋯xt+d.$$\begin{array}{} \displaystyle f_{t,d}(x_1,x_2,\ldots,x_{t}):=x_1x_2\cdots x_{t}+d. \end{array}$$ For any nonempty subset S of 𝔽, we say S  is  ft,d−closed  if  ft,d(x1,x2,…,xt):xi∈S and distinct⊆S.$$\begin{array}{} \displaystyle S ~~\text{is}~~ {f_{t,d}-closed} ~~\text{if}~~ \left\{f_{t,d}(x_1,x_2,\ldots,x_{t}):x_i\in S\text{ and distinct}\right\}\subseteq S. \end{array}$$ For any integer n , with t ≤ n ≤ |𝔽|, let 𝒰( n , t , d ) be the union of all f t , d -closed subsets S of 𝔽 with | S |= n . In this article, we investigate values of n , t , d for which 𝒰( n , t , d ) = 𝔽, with particular focus on t = n – 1, where n ∈ {3,4}. Moreover, if 𝒰( n , t , d )≠ 𝔽, we determine in many cases the exact elements of the set 𝔽∖ 𝔽( n , t , d ).

In this paper, we investigate some Cauchy problems involving a left-sided Hadamard-type fractional derivative. A theorem on the existence of a unique solution to a nonlinear problem is proved. The main result is obtained using a fixed point theorem due to Banach, as well as the Bielecki norm. A Cauchy formula for the solution of the linear problem is derived.

It is shown that for a function f : [ a , b ] × ℝ → ℝ which is measurable with respect to the first variable and upper semicontinuous quasicontinuous and increasing with respect to the second variable there exists a Caratheodory’s solution y(x)=y0+∫x0xf(t,y(t))dμ(t)$\begin{array}{} y(x)=y_0+\int\limits_{x_0}^xf(t,y(t)){\rm d}\mu(t) \end{array}$ of the Cauchy problem y ′( x ) = f ( x , y ( x )) with the initial condition y ( x 0 ) = y 0 . There is constructed an example which indicate to essentiality of condition of increasing and give the negative answer to a question of Z. Grande.

The celebrated Sturm-Picone comparison theorem as well as the well known Leighton’s variational lemma and Leighton’s theorem, all of which are the fundamental tools of comparison and so, oscillation theory, are obtained for regular and singular nonlinear impulsive differential equations and related previous results in the literature are generalized to such equations.

In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form (r(t)(y(t)+p(t)y(t−τ))″)″+q(t)G(y(t−σ))=0$$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption ∫0∞tr(t)dt=∞$$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$

In this paper, by using the concept of changing-periodic time scales and composition theorem of time scales introduced in 2015, we establish a local phase space for functional dynamic equations with infinite delay (FDEID) on an arbitrary time scale with a bounded graininess function μ . Through Krasnoseľskiĭ’s fixed point theorem, some sufficient conditions for the existence of local-periodic solutions for FDEID are established for the first time. This research indicates that one can extract a local-periodic solution for dynamic equations on an arbitrary time scale with a bounded graininess function μ through some index function.

Involutive Jamesian functions are functions aimed to predict the outcome of an athletic competition. They were introduced in 1981 by Bill James, but until recently little was known regarding their form. Using methods from quasigroup theory we are able to obtain a complete description of them.

Suppose that A , B ∈ 𝔹(𝓗) are positive invertible operators. In this paper, we show that A#B≤11−2μA12Fμ(A−12BA−12)A12 ≤12[A#B+Hμ(A,B)] ≤12[11−2μA12Fμ(A−12BA−12)A12+Hμ(A,B)] ≤⋯≤12nA#B+2n−12nHμ(A,B) ≤12n(1−2μ)A12Fμ(A−12BA−12)A12+2n−12nHμ(A,B) ≤12n+1A#B+2n+1−12n+1Hμ(A,B) ≤⋯≤Hμ(A,B)$$\begin{array}{} \displaystyle A \# B \leq \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}\\ \displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ A \# B +H_\mu (A,B)\bigg]\\ \displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+H_\mu (A,B)\bigg]\\ \displaystyle\qquad~\,\leq \dots \leq \frac{1}{2^n}A \# B + \frac{2^n-1}{2^n}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \frac{1}{2^n(1-2\mu)}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+\frac{2^n-1}{2^n}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \frac{1}{2^{n+1}} A \# B +\frac{2^{n+1}-1}{2^{n+1}}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \dots \leq H_\mu (A,B) \end{array}$$ for each μ∈[0,1]∖{12},$\begin{array}{} \displaystyle \mu \in [0,1]\smallsetminus\{\frac{1}{2}\}, \end{array}$ where H μ ( A , B ) and A # B are the Heinz mean and the geometric mean for operators A , B , respectively, and Fμ∈C(sp(A−12BA−12))$\begin{array}{} \displaystyle F_{\mu}\in C({\rm sp}(A^\frac{-1}{2}BA^\frac{-1}{2})) \end{array}$ is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.

In this paper, we improve the famous Reid inequality related to linear operators. Some monotony results for positive operators are also established with a different approach from what is known in the existing literature. Lastly, Reid’s (and Halmos-Reid’s) inequalities are extended to unbounded operators.

We introduce a notion of a probabilistic convergence transformation group, and present various natural examples including quotient probabilistic convergence transformation group. In doing so, we construct a probabilistic convergence structure on the group of homeomorphisms and look into a probabilistic convergence group that arises from probabilistic uniform convergence structure on function spaces. Given a probabilistic convergence space, and an arbitrary group, we construct a probabilistic convergence transformation group. Introducing a notion of a probabilistic metric convergence transformation group on a probabilistic metric space, we obtain in a natural way a probabilistic convergence transformation group.

Limit numbers (cluster sets) of a real function of a real variable were discussed in the literature by many authors. Generalizations of cluster sets were considered by distinctions of some classes of sets which generated some kind of limit. In general they were close to some topology on the set of real numbers. However not all such classes allowed to define a topology on ℝ in a simple way. We consider some topologies in ℝ generated by those classes of sets. We investigate a connection between limit numbers generated by those classes and limit numbers defined by a topology generated by a class 𝔅.

In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector defined by Lyberopoulos and Macheras (enlarging Huang’s original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial property within this class. In concrete examples, we demonstrate how to establish the Markov property by our results. Another consequence is the invariance of the Markov property under certain changes of measures.

In this paper, the author investigates the complete convergence for weighted sums of asymptotically negatively associated (ANA) random variables with different distributions, and obtains some equivalent conditions of complete convergence theorem for weighted sums as well as summation of ANA cases. These results generalize and improve the corresponding ones obtained by Baum and Katz (1965), Peligrad and Gut (1999) and Cai (2006), respectively.