# Scientific Journals and Yearbooks Published at SAS

## Article List

## Tatra Mountains Mathematical Publications

Volume 30, 2005, No. 1

Content:

- Jakubík, J.
**Generalized Boolean algebra extensions of lattice ordered groups.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 1-19. - Frič, R.
**Remarks on statistical maps and fuzzy (operational) random variables.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 21-34. - Černák, Š. - Lihová, J.
**Convergence with a regulator in lattice ordered groups.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 35-45. - Borsík, J.
**On strong quasicontinuity and continuity points.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 47-57. - Jakubíková-Studenovská, D.
**Cancellation law in class of monounary algebras.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 59-70. - Lihová, J.
**Convexities of Riesz groups.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 71-85. - Grande, Z.
**Convergence of sequences of functions having property $**$.*M*

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 87-92. - Purcz, P.
**Asymptotical behaviour of the speed-up of one parallel algorithm.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 93-100. - Repický, M.
**Good sequences for Sacks forcing.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 101-122. - Ploščica, M.
**Ideal lattices of locally matricial algebras.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 123-134. - Eliáš, J.
**Arbault permitted sets are perfectly meager.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 135-148. - Jendroľ, S. - Madaras, T.
**Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 149-153. - Mihók, P.
**On the lattice of additive hereditary properties of object systems.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 155-161. - Duchoň, M. - Haluška, J.
**On a generalized Kolmogoroff integral in complete bornological locally convex spaces.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 163-173. - Papčo, M.
**On fuzzy random variables: examples and generalizations.**

In*Tatra Mountains Mathematical Publications*. Vol. 30, no. 1 (2005), p. 175-185.

Generalized Boolean algebra extensions of lattice ordered groupsFulltext Ján Jakubík ^{1)}
To each pair ($A$, $B$), where $A$ is a lattice ordered group and $B$ is a generalized Boolean algebra, there corresponds a lattice ordered group $G$; the construction of $G$ is due to Conrad and Darnel. In this paper we deal with the relations between higher degrees of distributivity of the partially ordered structures $G$ and $B$. Further, we investigate direct product decomposition of $G$ in the case when $A$ is a linearly ordered group.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 1-19. | ||||||

Remarks on statistical maps and fuzzy (operational) random variablesFulltext Roman Frič ^{1)}
We study statistical maps, fuzzy random variables, observables, and some other basic notions of the fuzzy (operational) probability theory in the framework of a categorical duality theory for generallized measurable spaces and generalized fields of probability events.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 21-34. | ||||||

Convergence with a regulator in lattice ordered groupsFulltext Štefan Černák ^{1)}, Judita Lihová ^{2)}
The paper deals with Cauchy completions of archimedean lattice ordered groups with respect to a convergence of sequences with a fixed regulator.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 35-45. | ||||||

On strong quasicontinuity and continuity pointsFulltext Ján Borsík ^{1)}
The pair $(C(f), A(f))$, where $C(f)$ is the set of all continuity points and $A(f)$ is the set of all strong quasi-continuity points, is characterized. This is a solution of Problem 2 in [Z. Grande:
On strong quasicontinuity points, Tatra Mt. Math. Publ. 8 (1996), 17–21].Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 47-57. | ||||||

Cancellation law in class of monounary algebrasFulltext Danica Jakubíková-Studenovská ^{1)}
In this paper we deal with the validity of the implications $ABcong ACRightarrow Bcong C$ and $A
^{k}cong B^{k}Rightarrow Acong B$ $(kin Bbb N)$ for some classes of monounary algebras.Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 59-70. | ||||||

Convexities of Riesz groupsFulltext Judita Lihová ^{1)}
In this paper, the ordered class of all convexities of Riesz groups is investigated. Further, some principal convexities are dealt with.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 71-85. | ||||||

Convergence of sequences of functions having property $$MFulltext Zbigniew Grande ^{1)}
A function $f:{Bbb R} o {Bbb R}$ has the property $
$ if the restricted function $f/DM_{ap}(f)$ is continuous ($D_{ap}(f)$ denotes the set of all approximate discontinuity points of $f$). In this article I investigate the uniform, pointwise and transfinite limits of sequences of functions with the property $$.MTatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 87-92. | ||||||

Asymptotical behaviour of the speed-up of one parallel algorithmFulltext Pavol Purcz ^{1)}
An earlier suggested parallel ``ring'' algorithm for solving the spatially one-dimensional initial-boundary-value problem (IBVP) for a parabolic equation using an explicit difference method is shortly described. Asymptotical behaviour of the speed-up function of this parallel algorithm is studied. The speed-up function is determined as the ratio between necessary times for realization of the algorithm in sequentional and parallel cases. Theoretical estimates of the speed-up function show the significant speed-up of the parallel algorithm in comparison with the serial one for large values of the parameter $q$, where $q$ is the maximum of values computed by one processor during one time level. It is shown that the coefficient of the speed-up tends to number of using processors, if the parameter $q$ tends to infinity.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 93-100. | ||||||

Good sequences for Sacks forcingFulltext Miroslav Repický ^{1)}
We introduce an $ω$-closed partially ordered set $Bbb P
^{*}_{good}$ and prove that if it is $κ$-distributive then Sacks forcing is $(κ,frak c,ω)$-distributive. Moreover, we prove that abbr{PFA} implies that $Bbb P^{*}_{good}$ is $frak c$-distributive. We consider also some related partial orders, examine regularity properties for them, and find complete embeddings of the corresponding complete Boolean algebras.Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 101-122. | ||||||

Ideal lattices of locally matricial algebrasFulltext Miroslav Ploščica ^{1)}
We give another proof of Růžička's result that every infinite algebraic distributive lattice whose compact elements form a lattice is isomorphic to the lattice of all two-sided ideals of some locally matricial algebra. Our construction is more elementary and explicit.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 123-134. | ||||||

Arbault permitted sets are perfectly meagerFulltext Jozef Eliáš ^{1)}
We prove that every set permitted for the family of Arbault sets is perfectly meager. This negatively answers the question whether the existence of permitted sets of cardinality continuum is provable in ZFC.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 135-148. | ||||||

Note on an existence of small degree vertices with at most one big degree neighbour in planar graphsFulltext Stanislav Jendroľ ^{1)}, Tomáš Madaras ^{2)}
Let $G$ be a planar graph of minimum degree at least three that does not contain an edge with degree sum of its endvertices at most 8. We prove that every such graph $G$ contains a vertex of degree $d, d in {3,4,5}$, which has at least $d-1$ neighbours each of which has degree at most 20. The bound 20 is tight.
Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 149-153. | ||||||

On the lattice of additive hereditary properties of object systemsFulltext Peter Mihók ^{1)}
The notion of an
object-system over a concrete category ${C}$ is introduced as a common generalization of graphs, hypergraphs, digraphs and other mathematical structures. Let ${C}$ be a concrete category. A simple finite object-system over ${C}$ is an ordered pair $S = (V, E)$, where $V$ is a finite set and $E = {A_{1},A_{2},…, A_{m}}$ is a finite set of the objects of ${C}$, such that the ground-set $V(A_{i}) subseteq V$ of each object $A_{i} in E$ is a finite set. Analogously as for graphs, we define the additive hereditary property of simple object-systems as any class of systems closed under disjoint unions, subsystems and isomorphisms of systems, respectively. The structure of the lattice $Bbb L^{a}({C})$ of all additive hereditary properties of object-systems is investigated.Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 155-161. | ||||||

On a generalized Kolmogoroff integral in complete bornological locally convex spacesFulltext Miloslav Duchoň ^{1)}, Ján Haluška ^{2)}
We introduce a generalized Kolmogoroff integral of the first type with respect to the operator valued measure in complete bornological locally convex topological vector spaces and show that, in the equal setting, the class of integrable functions coincides with the class of integrable functions in the generalized Dobrakov integral sense, [J. Haluąka:
On integration in complete bornological locally convex spaces, Czechoslovak Math. J. 47, (1997), 205–219].Tatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 163-173. | ||||||

On fuzzy random variables: examples and generalizationsFulltext Martin Papčo ^{1)}
There are random experiments in which the notion of a classical random variable, as a map sending each elementary event to a real number, does not capture their nature. This leads to fuzzy random variables in the Bugajski–Gudder sense. The idea is to admit variables sending the set $Ω$ of elementary events not into the real numbers, but into the set $M
_{1}^{+}(Bbb R)$ of all probability measures on the real Borel sets (each real number $rin Bbb R$ is considered as the degenerated probability measure $δ_{r}$ concentrated at $r$). We start with four examples of random experiments ($Ω$ is finite); the last one is more complex, it generalizes the previous three, and it leads to a general model. A fuzzy random variable is a map $φ$ of $M_{1}^{+}(Ω)$ into $M_{1}^{+}(Ξ)$, where $M_{1}^{+}(Ξ)$ is the set of all probability measures on another measurable space $(Ξ,(Ξ))$, satisfying certain measurability condition. We show that for discrete spaces the measurability condition holds true. We continue in our effort to develop a suitable theory of $ID$-posets, $ID$-random variables, and $ID$-observables. Fuzzy random variables and Markov kernels become special cases.BTatra Mountains Mathematical Publications. Volume 30, 2005, No. 1: 175-185. |