# Mathematica Moravica, Vol. 4 (2000)

Correction d'une Demonstration dans un Travail Anterieur
Mathematica Moravica, Vol. 4 (2000), 1–4.

Abstract. Il s'agit dans cette Note de la correction d'une partie de la démonstration du Lemme 1 dans le trafail [1], suivie de quelques considérations supplémentaires.
Keywords. Équation functionelle, espace linéaire, base d'Hamel.

On p-Semigroups
Mathematica Moravica, Vol. 4 (2000), 5–20.

Abstract. Generalizing the notion of anti-inverse semigroup, we introduce the notion of $p$-semigroup, for arbitrary $p\in N$. We prove that every $p$-semigroup is covered by groups, clases of which are completely described.
Keywords. Semigroup, union of groups, $p$-semigroups.

An Estimation of Approximation for the Solution of Ordinary Differential Equations
Mathematica Moravica, Vol. 4 (2000), 21–26.

Abstract. In this paper we give an estimation of the method for finding approximative solution of ordinary differential equations of the first order and these equations systems by use of operators which satisfy condition $|(U(\psi_{1}))(x) − (U(\psi_{2}))(x)| \leq A_{0}\|\psi_{1} - \psi_{2}\|_{C}.$ Keywords. Ordinary differential equations, approximative solution, operationale equation, system of ordinary differential equations.

Initial Segments in BCC-algebras
Mathematica Moravica, Vol. 4 (2000), 27–34.

Abstract. The role of initial segments in BCC-algebras is described.
Keywords. BCC-algebra, BCK-algebra, atom.

A Convergence Criterion of Series
Mathematica Moravica, Vol. 4 (2000), 35–37.

Abstract. In this article is given a convergence criterion of series, which is similar to classical criterions of Abel and Dirichlet, and also some of its consequences and applications.
Keywords. Convergence criterion of series.

The Prelimit of a Real-Valued Function
Mathematica Moravica, Vol. 4 (2000), 39–44.

Abstract. 1. In [1] S. Banach shown the existence of very known Banach linear shift-invariant functionals defined on the real vector space of all bounded real-valued functions on the semi-axis t≥0 and especially on the space of all real bounded sequences. In [2] G.G. Lorentz defined, by Banach shift-invariant functionals, the class of almost convergent sequences. In [3] almost convergence was extended to real-valued functions on the semi-axis $t\geq 0$. In [4] almost convergance was extended to bounded sequences in a real normed space.
2. This paper is devoted to a class of functions defined on the semi-axis $t\geq 0$, which are near to the functions $f$ having $lim_{t\to \infty}f(t)$. The paper is organized as follows. First, for a sufficiently large $a$ (written $a > a_0$ for some $a_0$) by $\Omega$ we denote the real vector space of all functions defined on $[0,+\infty)$. Next, we will show the existence of a family of functionals defined on the space $\Omega$. By these functionals we define the notion of a function $f\in\Omega$ and investigate the family of all these functions. Further, we will show a theorem characterizing a function having a pre-limit. Also, we show another theorem which is very applicable, though it contains a new restrictive condition. Finally, to make the idea of pre-limit a little clearer, we give several examples functions having pre-limit.
Keywords. Functional, Pre-limit, Convergence, Pre-convergent.

Completeness Theorem for Boolean Models with Strictly Positive Measure
Mathematica Moravica, Vol. 4 (2000), 45–50.

Abstract. Rašković [3] introduce a conservative extension of classical propositional logic with some probability operators and prove corresponding completeness and decidability theorem. The aim of this paper is to prove Robinson consistency and Craig interpolation for this logic.
Keywords. Probability logic, Strictly positive measure, Booleand models.

On Eigenvalues and Main Eigenvalues of a Graph
Mathematica Moravica, Vol. 4 (2000), 51–58.

Abstract. Let $G$ be a simple graph of order $n$ and let $\lambda_{1}\geq \lambda_{2}\geq \cdots \geq\lambda_{n}$ and $\lambda_{1}^{\ast}\geq\lambda_{2}^{\ast}\cdots\lambda_{n}^{\ast}.$ be its eignevalues with respect to the ordinary adjacency matrix $A=A(G)$ and the Seidel adjacency matrix $A^{\ast}=A^{\ast}(G)$, respectively. Using the Courant-Weyl inequalities we prove that $\bar{\lambda}_{n+1-i}\in [-\lambda_{i}-1,-\lambda_{i+1}-1]$ and $\lambda_{n+1-i}^{\ast}\in [-2\lambda_{i}-1,-2\lambda_{i+1}-1]$ for $i=1,2,\dots,n-1$, where $\bar{\lambda}_{i}$ are the eigenvalues of its complement $\bar{G}$. Besides, if $G$ and $H$ are two switching equivalent graphs, the we find $\lambda_{i}(G)\in[\lambda_{i+1}(H),\lambda_{i-1}(H)]$ for $i=2,3,\dots,n-1$. Next, let $\mu_{1},\mu_{2},\dots,\mu_{k}$ and $\bar{\mu}_{1},\bar{\mu}_{2},\dots,\bar{\mu}_{k}$ denote the main eigenvalues of the graph $G$ and the complementary graph $\bar{G}$, respectively. In this paper we also prove $\sum_{i=1}^{k}(\mu_{i}+\bar{mu}_{i}) = n-k$.
Keywords. Graph, eigenvalue, main eigenvalue.

Submeasures with Probabilistic Structures
Mathematica Moravica, Vol. 4 (2000), 59–65.

Abstract. In [8] the author gives some probabilistic generalizations of the submeasure concept. The purpose of this paper is to define a general form of submeasure with probabilistic structure in wuch a way that the topological ring of sets is a uniform space. As particular cases the probabilistic generalizations from [8] are obtained.
Keywords. Probabilistic structure, uniform space.

Some Properties Similar to Countable Compactness and Lindelöf Property
Mathematica Moravica, Vol. 4 (2000), 67–73.

Abstract. In this paper we further investigate the results given in [7, 8, 9]. In Section 2 we consider space $X$ for which the closure of each countably compact (strongly countably compact, hypercountably compact) subspace of $X$ has countably compact (strongly countably compact, hypercountablycompact) property. In Section 3 we study some notions related to the classical concepts of being a Lindelöf, Menger or a Rothberger space.
Keywords. Compactness and Lindelöf property.

Some Properties of Spaces Similar to Čech-Complete Property
Mathematica Moravica, Vol. 4 (2000), 75–82.

Abstract. In this paper we study some notions related to the remainder $X^{\ast}=\beta X\backslash\beta(X)$ which are similar to the Čech-complete property. A topological space $X$ is $P(\omega P)$-complete if $X$ is a Tychonoff space and remainder $X^{\ast}=\beta X\backslash\beta(X)$ is a $P(\omega P)$-set in $\beta X$. The set $A\subset X$ is an $L$-set if $A\cap cl_{X}(F)=\emptyset$ for each Lindelöf subset $F$ contained in $X\backslash A$. Recall that a space $X$ is said to be $L$-complete if $X$ is a Tychonoff space and the remainder $X^{\ast}=\beta X\backslash\beta(X)$ is and $L$-set in $\beta X$.
Keywords. Extension spaces, remainders, completions.

Some Characterization of Lorentzian Spherical Spacelike Curves with the Timelike and the Null Principal Normal
Mathematica Moravica, Vol. 4 (2000), 83–92.

Abstract. In [4] the authors have characterized Lorentzian spherical sapcelike curves in the 3-dimensional Minkowski space with the spacelike normal. In this paper, we shall characterize the Lorentzian spherical spacelike curves in the same space with the timelike and the null principal normal.
Keywords. Lorentzian sphere, spacelike curve, frenet formula, Minkowski space, Lorentzian inner product.

Some Fixed Point Theorems for Multi-Valued Mappings on Reflexive Banach Spaces
Mathematica Moravica, Vol. 4 (2000), 93–98.

Abstract. Some fixed point theorems for multi-valued mappings satisfying an implicit relation which generalize the main results from [1] are proved.
Keywords. Multi-valued mapping, common fixed point, implicit relation, reflexive Banach space.

A Group Blind Signature Scheme Based on the Strong RSA Assumption
Mathematica Moravica, Vol. 4 (2000), 99–108.

Abstract. A group blind signature require that a group member signs on group's behalf a document wthout knowing its content. In this paper we propose an efficient and provably secure group blind signature scheme. Our scheme is an extension of Camernisch and Michels' group signature scheme [2] that adds the blindness property. The proposed group blind signature scheme is more efficient and secure than Lysyanskaya-Ramzan scheme [12].
Keywords. Group blind signature scheme, group signatures, blind signatures.

A Note on the Post's Coset Theorem
Mathematica Moravica, Vol. 4 (2000), 109–114.

Abstract. In this paper a proff o Post's Coset Theorem is presented. The proof uses from Theory of $n$-groups, besides the definition of $n$-groups ([1], 1.1), the description of $n$-group as an algebra with the laws of the type $\langle n,n-1,n-1\rangle$ ([8], 1.2, 1.3).
Keywords. $n$-groupoids, $n$-semigroups, $n$-quasigroups, $n$-groups, $\{1,n\}$-neutral operations on $n$-groupoids, inversing operation on $n$-group.
On $(n,m)$-Groups
Abstract. The main result of the article is the following proposition. Let $n\geq 2m$ and let $(Q,A)$ be an $(n,m)$-groupoid. Then, $(Q,A)$ is an $(n,m)$-group iff the following statements hold: $(i)$ $(Q,A)$ is an $\langle 1,n-m+1\rangle$- and $\langle 1,2\rangle$-associative $(n,m)$-groupoid [or $\langle 1,n-m+1\rangle$- and $\langle n-m,n-m+1\rangle$-associative $(n,m)$-groupoid]; and $(ii)$ for every $a_{1}^{n}\in Q$ there is at least one $x_{1}^{m}\in Q^{m}$ and at least one $y_{1}^{m}\in Q^{m}$ such that the following equalities hold $A(a_{1}^{n-m},x_{1}^{m}) = a_{n-m+1}^{n}$ and $A(y_{1}^{m},a_{1}^{n-m})=a_{n-m+1}^{n}$. [For $n=2$ and $m=1$ it is a well known characterization of a group. See, also 3.2]
Keywords. $(n,m)$-groupoids, $(n,m)$-groups.