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Tatra Mountains Mathematical Publications


Volume 30, 2005, No. 1

Content:


 
Generalized Boolean algebra extensions of lattice ordered groups.

Ján Jakubík 1)

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1)Matematický ústav SAV, Grešákova 6; 040 01 Košice.

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.

How to cite (APA format):
Jakubík, J. (2005). Generalized Boolean algebra extensions of lattice ordered groups. Tatra Mountains Mathematical Publications, 30(1), 1-19.


 
 
Remarks on statistical maps and fuzzy (operational) random variables.

Roman Frič 1)

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1)Matematický ústav SAV, Grešákova 6; 040 01 Košice.

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.

How to cite (APA format):
Frič, R. (2005). Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mountains Mathematical Publications, 30(1), 21-34.


 
 
Convergence with a regulator in lattice ordered groups.

Štefan Černák 1), Judita Lihová 2)

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1)Katedra matematiky, Stavebná fakulta TU; Vysokoškolská 4; 042 02 Košice.
2)Katedra geometrie a algebry, PF UPJ©; Jesenná 5; 041 54 Koąice.

The paper deals with Cauchy completions of archimedean lattice ordered groups with respect to a convergence of sequences with a fixed regulator.

How to cite (APA format):
Černák, Š, Lihová, J. (2005). Convergence with a regulator in lattice ordered groups. Tatra Mountains Mathematical Publications, 30(1), 35-45.


 
 
On strong quasicontinuity and continuity points.

Ján Borsík 1)

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1)Matematický ústav SAV, Grešákova 6; 040 01 Košice; Slovensko. borsik@saske.sk

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].

How to cite (APA format):
Borsík, J. (2005). On strong quasicontinuity and continuity points. Tatra Mountains Mathematical Publications, 30(1), 47-57.


 
 
Cancellation law in class of monounary algebras.

Danica Jakubíková-Studenovská 1)

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1)Katedra geometrie a algebry, PF UPJ; Jesenná 5; 041 54 Košice.

In this paper we deal with the validity of the implications $ABcong ACRightarrow Bcong C$ and $Akcong BkRightarrow Acong B$ $(kin Bbb N)$ for some classes of monounary algebras.

How to cite (APA format):
Jakubíková-Studenovská, D. (2005). Cancellation law in class of monounary algebras. Tatra Mountains Mathematical Publications, 30(1), 59-70.


 
 
Convexities of Riesz groups.

Judita Lihová 1)

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1)Katedra geometrie a algebry, PF UPJ©; Jesenná 5; 041 54 Koąice.

In this paper, the ordered class of all convexities of Riesz groups is investigated. Further, some principal convexities are dealt with.

How to cite (APA format):
Lihová, J. (2005). Convexities of Riesz groups. Tatra Mountains Mathematical Publications, 30(1), 71-85.


 
 
Convergence of sequences of functions having property $M$.

Zbigniew Grande 1)

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1)Department of Mathematics Pedagogical University, Plac Slowianski 9, 65-069 Zielona Gora, POLAND. grande@ukw.edu.pl.

A function $f:{Bbb R} o {Bbb R}$ has the property $M$ if the restricted function $f/Dap(f)$ is continuous ($Dap(f)$ 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 $M$.

How to cite (APA format):
Grande, Z. (2005). Convergence of sequences of functions having property $M$. Tatra Mountains Mathematical Publications, 30(1), 87-92.


 
 
Asymptotical behaviour of the speed-up of one parallel algorithm.

Pavol Purcz 1)

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1)% Katedra matematiky, % SvF TU v Košiciach; % Vysokoškolská 4; % 042 00 KOŠICE %.

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.

How to cite (APA format):
Purcz, P. (2005). Asymptotical behaviour of the speed-up of one parallel algorithm. Tatra Mountains Mathematical Publications, 30(1), 93-100.


 
 
Good sequences for Sacks forcing.

Miroslav Repický 1)

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1)Matematický ústav SAV, Jesenná 5; 041 54 Košice; Slovensko. repicky@kosice.upjs.sk

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.

How to cite (APA format):
Repický, M. (2005). Good sequences for Sacks forcing. Tatra Mountains Mathematical Publications, 30(1), 101-122.


 
 
Ideal lattices of locally matricial algebras.

Miroslav Ploščica 1)

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1)Matematický ústav SAV, Grešákova 6; 040 01 Košice.

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.

How to cite (APA format):
Ploščica, M. (2005). Ideal lattices of locally matricial algebras. Tatra Mountains Mathematical Publications, 30(1), 123-134.


 
 
Arbault permitted sets are perfectly meager.

Jozef Eliáš 1)

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1)Katedra matematiky, FHI EU; Dolnozemská cesta 1; 852 35 Bratislava.

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.

How to cite (APA format):
Eliáš, J. (2005). Arbault permitted sets are perfectly meager. Tatra Mountains Mathematical Publications, 30(1), 135-148.


 
 
Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs.

Stanislav Jendroľ 1), Tomáš Madaras 2)

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1)Katedra geometrie a algebry, UPJŠ; Jesenná 5; 041 54 Košice, Slovenská republika. jendrol@upjs.sk
2)Katedra geometrie a algebry, Univerzita P. J Šafárika; Jesenná 5; Košice; Slovenská reublika. tomas.madaras@upjs.sk

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.

How to cite (APA format):
Jendroľ, S, Madaras, T. (2005). Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs. Tatra Mountains Mathematical Publications, 30(1), 149-153.


 
 
On the lattice of additive hereditary properties of object systems.

Peter Mihók 1)

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1)Matematický ústav SAV, Grešákova 6; 040 01 Košice.

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 $E = {A1,A2,…, Am}$ is a finite set of the objects of ${C}$, such that the ground-set $V(Ai) subseteq V$ of each object $Ai in E$ 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 $Bbb La({C})$ of all additive hereditary properties of object-systems is investigated.

How to cite (APA format):
Mihók, P. (2005). On the lattice of additive hereditary properties of object systems. Tatra Mountains Mathematical Publications, 30(1), 155-161.


 
 
On a generalized Kolmogoroff integral in complete bornological locally convex spaces.

Miloslav Duchoň 1), Ján Haluška 2)

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1)Matematický ústav SAV, Štefánikova 49; 814 73 Bratislava; SLOVENSKO. duchon@mat.savba.sk
2)Matematický ústav SAV, Grešákova 6; 040 01 Košice. jhaluska@mail.saske.sk

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].

How to cite (APA format):
Duchoň, M, Haluška, J. (2005). On a generalized Kolmogoroff integral in complete bornological locally convex spaces. Tatra Mountains Mathematical Publications, 30(1), 163-173.


 
 
On fuzzy random variables: examples and generalizations.

Martin Papčo 1)

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1)Matematický ústav SAV, Grešákova 6; 040 01 Košice; Slovensko.

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 $M1+(Bbb R)$ of all probability measures on the real Borel sets (each real number $rin Bbb R$ is considered as the degenerated probability measure $δr$ 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 $M1+(Ω)$ into $M1+(Ξ)$, where $M1+(Ξ)$ is the set of all probability measures on another measurable space $(Ξ,B(Ξ))$, 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.

How to cite (APA format):
Papčo, M. (2005). On fuzzy random variables: examples and generalizations. Tatra Mountains Mathematical Publications, 30(1), 175-185.