An Altman Type Generalization of the Brézis-Browder Ordering Principle

Mathematica Moravica, Vol. **5** (2001), 1–6.

doi: http://dx.doi.org/10.5937/MatMor0105001S

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**Abstract.** By making use the ideas of M. Altman, we prove a more natural
generalization of the famous ordering principle of H. Brézis and F.E. Browder.

**Keywords.** Preodered sets, monotonicity and boundedness properties, existence of maximal elements.

Primitive Idempotents in Semigroups

Mathematica Moravica, Vol. **5** (2001), 7–18.

doi: http://dx.doi.org/10.5937/MatMor0105007B

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**Abstract.** In this paper we study certain general properties of primitive and 0-primitive idempotents,
and the obtained results we apply to 0-inversive E-inversive and B-inversive semi-groups. We also determine some conditions under which
a left ideal of a semi-group with zero generated by a nonzero idempotent is left 0-simple. The obtained results generalize some results by Venkatesan,
Steinfeld, Bogdanović and Milić, Bogdanović and Ćirić, Mitsch and Petrich and others.

**Keywords.** Idempotents, primitive idempotents, semi-groups, E-inversive semi-groups, B-inversive semigroups, ideals, minimal ideals.

Tauberian Theorems for Convergence and Subsequential Convergence with Moderately Oscillatory Behavior

Mathematica Moravica, Vol. **5** (2001), 19–56.

doi: http://dx.doi.org/10.5937/MatMor0105019D

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**Abstract.** In the classical Tauberian theory, the main objective is to obtain convergence of
a sequence $\{u_n\}$ by imposing conditions about the oscillatory behavior of $\{u_n\}$ in addition to the existence of certain continuous limits.
However, there are some conditions of considerable interest from which it is not possible to obtain convergence of $\{u_n\}$.
This situation motivates a different kind of Tauberian theory where we do not look for convergence recovery of $\{u_n\}$,
rather we are concerned with the subsequential behavior of the sequence $\{u_n\}$. The first section includes definitions,
notations and an overview of classical results. Succinct proofs of the Hardy-Littlewood theorem and the generalized
Littlewood theorem are given using the corollary to Karamata's Hauptsatz. In the second section subsequential
Tauberian theory is introduced and some related Tauberian theorems are proved.
Finally, in the last section we study convergence and subsequential convergence of regularly generated sequences.

**Keywords.** Tauberian theorems, subsequential Tauberian theorems.

Tauberian Theorems for Sequences with Moderately Oscillatory Control Modulo

Mathematica Moravica, Vol. **5** (2001), 57–94.

doi: http://dx.doi.org/10.5937/MatMor0105057D

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**Abstract.** We introduce a general control modulo of the oscillatory behavior of order $m$ of $\{u_n\}$,
which leads to new Tauberian conditions and consequently to the new Tauberian theorems. Also the notion of moderately oscillatory and
regularly generated sequences is presented and studied. In the first section we give basic definitions, notations and a brief survey of classical results.
Next we establish Tauberian theorems by using the general control modulo.
The proofs of these theorems are based on the classical and neoclassical Tauberian results, in particular on the corollary to Karamata's Hauptsatz.
Finally in the last section we consider the class of moderately oscillatory regularly generated sequences and prove some theorems similar to Tauberian theorems.

**Keywords.** Tauberian theorems for limiting processes with controlled and managed oscillatory behavior.

Roughly $d$-Convex Functions on Undirected Tree Networks

Mathematica Moravica, Vol. **5** (2001), 95–101.

doi: http://dx.doi.org/10.5937/MatMor0105095M

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**Abstract.** In this paper we establish some properties of roughly $d$-convex functions on
undirected tree networks.
It is pointed out that these roughly $d$-convex functions have the following properties concerning the property of minimum each local minimum of
a midpoint $\delta$-$d$-convex or lightly $\gamma$-$d$-convex functions is a global minimum, where a local with radius equal to the roughness degree.
Since every $p$-$d$-convex function is midpoint $\gamma$-$d$-convex and every $\gamma$-$d$-convex function is lightly $\gamma$-$d$ convex,
this conclusion holds for them, too. We also state weaker but sufficient conditions for roughly $d$-convex functions.
We adopt the definition of network as metric space introduced by Dearing P.M. and Francis R.L. in 1974.

**Keywords.** $p$-$d$-convex, $b$-$d$-convex, midpoint $b$-$d$-convex, $\gamma$-$d$-convex, lightly $\gamma$-$d$-convex,
midpoint $\gamma$-$d$-convex, strictly $\gamma$-$d$-convex, strictly $r$-$d$-convexlike functions, tree networks.

Dynamics on $(P_{cp}(X), H_{d})$ Generated by a Finite Family of Multi-valued Operators on $(X, d)$

Mathematica Moravica, Vol. **5** (2001), 103–110.

doi: http://dx.doi.org/10.5937/MatMor0105103P

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**Abstract.** The main purpose of this paper is to give some partial answers to the following problem:

If $F_i$, $i\in\{1,\dots,m\}$ is a finite family of weakly Picard multi-valued operators, is the operator

$T_{F}: P(X) \to P(X)$, $T_{F}:= \bigcup_{i=1}^{m} F_{i}(Y)$
a weakly Picard operator too.

**Keywords.** Weakly Picard operator, Hausdorf-Pompeiu generalized functional, generalized contraction.

Fixed Points and Apices on Arbitrary Sets

Mathematica Moravica, Vol. **5** (2001), 111–118.

doi: http://dx.doi.org/10.5937/MatMor0105111T

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**Abstract.** This paper presents new statements for fixed points and apices arbitrary nonempty sets.
Applications in fixed point theory and for general quasi-metric spaces are considered.

**Keywords.** Fixed points, fixed apex, fixed apices, periodic points, fixed pointson arbitrary sets, general quasi-metric spaces.

On Monotone Polynomial Interpolation

Mathematica Moravica, Vol. **5** (2001), 119–128.

doi: http://dx.doi.org/10.5937/MatMor0105119T

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**Abstract.** An older demonstration of the existence of an interpolatory piecewise monotone function is revisited.
A new strategy to decrease the degree of interpolation polynomial is presented.

**Keywords.** Monotone polynomial interpolation.

Description of Super Associative Algebras With $n$-Quasigroup Operations

Mathematica Moravica, Vol. **5** (2001), 129–157.

doi: http://dx.doi.org/10.5937/MatMor0105129U

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**Abstract.** Let $\Sigma$ be a set operation over $Q$.
Let also $w_{1} = w_{2}$ be a law in a description of which variables $x_{1},\dots,x_{s}$ are included, and also operational symbols
$X_{1},\dots,X_{k}$, whose set of lengths is a subset of the set of lengths of operations from $\Sigma$.
Then $(Q, \Sigma)$ is said to be an algebra with the super identity $w_{1} = w_{2}$ if for every substitution of the variables $x_{1},\dots,x_{s}$
with elements of $Q$ and for every substitution of operational symbols $X_{1},\dots,X_{k}$ with operations from
$\Sigma$ [with the corresponding lengths] $w_{1} = w_{2}$ becomes an equality in $(Q, \Sigma)$; [2].
Quasigroup algebras with associative superlaws were described by V.D. Belousov [5] (See also [16]).
3-quasigroup algebras with associative superlaws were primary described by Yu. M. Movsisyan ([9], p. 152-158).
(Associative superlaws of hyperidentities of associativity; see also [15])
In the present paper, for $n$-quasigroup algebras with associative superlaws, the author was free to use the name:
super associative algebras of $n$-quasigroup operations [briefly: $SAAnQ$].
In the paper, primary, in a unique way are described nontrivial $SAAnQ$ [briefly:
$NetSAAnQ$] for every $n\in N\backslash\{1\}$ with an exception of a case for $n = 2$.
The crucial role in the mentioned description of $NetSAAnQ$ play the $\{1, n\}$-neutral and the inversing operations in an $n$-group.
Starting with the mentioned description of $NetSAAnQ$, these algebras for $n\geq 3$ are finally described in terms of Hosszú-Gluskin algebras of order $n$.

**Keywords.** $n$-semigroups, $n$-quasigroups, $n$-groups, $\{1, n\}$-neutral operations on $n$-groupoids,
inversing operation on $n$-group, central operation on $n$-group, $nHG$-algebras.

A Comment on $(n, m)$–Groups for $n\geq 3m$

Mathematica Moravica, Vol. **5** (2001), 159–162.

doi: http://dx.doi.org/10.5937/MatMor0105159U

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**Abstract.** In the present paper the following proposition is proved.
Let $n\geq 3m$ and let $(Q,A)$ be an $(n, m)$-groupoid.
Then, $(Q,A)$ is an $(n, m)$-group if for some $i\in \{m + 1,\dots, n − 2m + 1\}$ the following conditions hold:
(a) the $\langle i − 1,i\rangle$-associative law holds in $(Q,A)$; (b) the $\langle i, i + 1\rangle$-associative law holds in $(Q,A)$; and
(c) for every $a_{1}^{n}\in Q$ there is exactly one $x_{1}^{m}\in Q$ such that the following equality holds
$A(a_{1}^{i-1}, x_{1}^{m}, a_{i}^{n−m}) = a_{n-m+1}^{n}$.

**Keywords.** $(n, m)$-groupoids, $(n, m)$-semigroupoids, $(n, m)$-semigroup, $(n; m)$-group.

On a Family of $(n+1)$–ary Equivalence Relations

Mathematica Moravica, Vol. **5** (2001), 163–167.

doi: http://dx.doi.org/10.5937/MatMor0105163U

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**Abstract.** The notation of a partition of type $n$, $(n\in N)$, was introduced by J. Hartmanis in [1]
as a generalization of the notion of an ordinary partition of a set. It is well-known fact that partitions $Q$ (of type 1) correspond
in an one-one way to equivalence relations on $Q$. In this article we introduce an analogous family of relations $(\mathcal{F}_{n}(Q))$ for partitions of type $n$.
Furthermore for $p\in \mathcal{F}_{n}(Q)$ the following statements hold: $○(\overset{n+1}{\rho}) = \rho$ and $(\overset{n}{\rho})^{-1} = \rho$
for $n = 1: \sim ○ \sim = \sim$ and $\sim$ (cf. [3]). A similar family of relations for partitions of type $n$
was described by H.E. Pickett in [2] point out the differences.

**Keywords.** Partitions of type $n$, $(n + 1)$-ary equivalence relation.