# Mathematica Moravica, Vol. 10 (2006)

Cardinal Invariants for Commutative Group Algebras
Mathematica Moravica, Vol. 10 (2006), 1–7.

Abstract. A new kind of major structural invariants for commutative group algebras and pairs of commutative group algebras are here obtained. The present statements are a sequel to our recent results published in Ricerche Math. (Napoli, 2001 and 2003) plus Rend. Circolo Mat. Palermo (2002).
Keywords. Group algebras, purifiable subgroups, pure hulls, defects, covering dimensions, complemental dimensions, Hill’s numbers.

On a Generalization of Normal, Almost Normal and Mildly Normal Spaces
Mathematica Moravica, Vol. 10 (2006), 9–20.

Abstract. In this paper, we introduce the notions of $\delta p$-normal spaces, almost $\delta p$-normal spaces, mildly $\delta p$-normal spaces, $g\delta p$-closed sets and the forms of generalized $\delta$-preclosed functions. We obtain characterizations and the relationships of such normal spaces, properties of the forms of generalized $\delta$-preclosed functions and preservation theorems.
Keywords. $\delta p$-normal space, almost $\delta p$-normal space, mildly $\delta p$-normal space, $g\delta p$-closed set, generalized function.

One Observation on (n,m)-Semigroups
Mathematica Moravica, Vol. 10 (2006), 21–26.

Abstract. In this paper one Čupona-Trpenovski’s theorem about $n$-semigroups with neutral element is generalized.
Keywords. $n$-semigroup; $(n,m)$-semigroup; $n$-group; $(n,m)$-group.

Fixed Point Theorems for Two Pairs of Nonself Mappings in Metrically Convex Spaces by Altering Distances
Mathematica Moravica, Vol. 10 (2006), 27–40.

Abstract. Some common fixed point theorems for two pairs of nonself mappings in complete metrically convex metric spaces are proved by altering distances between the points which generalize earlier results due to Khan et al. [15] Khan and Bharadwaj [14], Bianchini [5], Chatterjea [6] and others. Some related results are also discussed besides furnishing an illustrative example.
Keywords. Metrically convex metric space, weak commutativity, compatible mappings, coincidentally commuting mappings, complete metric spaces.

Fixed Point Theorems in Banach Spaces Over Topological Semifields
Mathematica Moravica, Vol. 10 (2006), 41–46.

Abstract. A fixed point theorem for five pair wise commuting mappings on a Banach space $X$ over a topological semifield is proved which improves and extends the main result of [5]. In the sequel, an application is given for the solvability of certain non-linear functional equation in $X$.
Keywords. Banach space, common fixed points, topological semifield.

Intuitionistic Fuzzy Strongly Preopen (Preclosed) Mappings
Mathematica Moravica, Vol. 10 (2006), 47–53.

Abstract. Intuitionistic fuzzy strongly preopen (preclosed) mappings between intuitionistic fuzzy topological spaces are introduced. Some of their properties are studied.
Keywords. Intuitionistic fuzzy topological space, intuitionistic fuzzy strongly preopen set, intuitionistic fuzzy strong precontinuity, intuitionistic fuzzy strongly preopen mapping, intuitionistic fuzzy strongly preclosed mapping.

A Common Fixed Point Theorem for Multivalued Mappings Through T-weak Commutativity
Mathematica Moravica, Vol. 10 (2006), 55–60.

Abstract. In this paper, we prove a common fixed point theorem for a single-valued and the multivalued mappings by using $T$-weak commutativity condition. We also show that continuity of any mapping is not needed for the existence of the common fixed point.
Keywords. Common fixed point, $T$-weakly commuting mappings.

A Theorem About Uniformly Converging Progression of Constructive Functions
Mathematica Moravica, Vol. 10 (2006), 61–65.

Abstract. We give an example which represents an application of constructive functions in mathematical analysis, especially, to the theory of progressions. In fact we shall prove a theorem related to the uniformly converging progression of constructive functions whose sum is not $R$-integrable on the unit interval.
Keywords. Uniformly converging, constructive functions, theory of progressions, not integrable, $R$-integral, algorithms, normal algorithms, quasi-numbers, quasi-integrable.

Fixed Point Theorems for Hybrid Mappings in Metric Spaces
Mathematica Moravica, Vol. 10 (2006), 67–76.

Abstract. Some common fixed point theorems for non-self hybrid mappings have been proved by altering the distance between the points. Our results extends and generalizes many well known results.
Keywords. Hybrid Mapping, contractive type mappings, compatible mappings, common fixed points, Multivalued mappings.

Distributive Lattices and Hurewicz Families
Mathematica Moravica, Vol. 10 (2006), 77–90.

Abstract. Hurewicz and Rothberger respectively introduced prototypes of the selection properties $\mathsf{S}_{fin}(\mathcal{A},\mathcal{B})$ and $\mathsf{S}_{1}(\mathcal{A},\mathcal{B})$. In the series of papers titled “Combinatorics of open covers” (see the bibliography) we learned that for various topologically significant families $\mathcal{A}$ and $\mathcal{B}$ these selection properties are intimately related to game theory and Ramsey theory. The similarity in techniques used there to explore these relationships suggests that there should be a general, unified way to obtain these results. In this paper we pursue one possibility by considering the selection principle $\mathsf{S}_{fin}(\mathcal{A},\mathcal{B})$ for distributive lattices. The selection principle $\mathsf{S}_{1}(\mathcal{A},\mathcal{B})$ for distributive lattices will be treated in [3]. We use two examples throughout to illustrate the generality of the methods developed here.
Keywords. Game, selection principle, lattice, $\mathcal{A}$-tree, Hurewicz $\mathcal{A}$-tree, Hurewicz pairs, Ramsey family, partition relation, selectable pair.

On Super-Associative Algebras with $(2m,m)$-Quasigroup Operations for $m\geq 2$
Mathematica Moravica, Vol. 10 (2006), 91–100.

Abstract. In this paper super-associative algebras with $(2m,m)$-quasigroup operations are considered. Case $m = 1$ is described in [1]. Super-associative algebras with $n$-quasigroup operations for $n = 3$ Yu. Movsisyan was described in 1984 (cf. [6]). Case $n\geq 3$ was described in [10]. See, also [11].
Keywords. $(n,m)$-groupoid, $(n,m)$-group, $n$-group, $\{1, n-m+1\}$-neutral operation of the $(n,m)$-groupoid.

n-Groups $(n>3)$ as Algebras of the Type $\langle n,n-2,n-2\rangle$ with Four Laws
Mathematica Moravica, Vol. 10 (2006), 101–106.

Abstract. In this article $n$-groups $(n > 3)$ are described as algebras of the type $\langle n,n - 2,n - 2\rangle$ with four laws.
Keywords. $n$-semigroup, $n$-quasigroup, $n$-group, $\{i,j\}$-neutral operation on $n$-groupoid.

$(n,m)$-Groups in the Light of the Neutral Operations
Mathematica Moravica, Vol. 10 (2006), 107–147.

Abstract. This text is as an attempt to systematize the results about $(n,m)$-groups in the light of the neutral operations. (The case $m = 1$ is the monograph [23]).
Keywords. $(n,m)$-group, $n$-group, $\{1, n-m+1\}$-neutral operation of the $(n,m)$-groupoid.
A Survey of $NP$-Polyagroups (Survey article)
Keywords. $n$-group, $\{1, n\}$-neutral operation of the $n$-groupoid, near-$P$-polyagroup of the type $(s, n-1)$.