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SCIENTIFIC ISSUES
JAN DŁUGOSZ UNIVERSITY
in CZĘSTOCHOWA
MATHEMATICS XXIII
Scientiﬁc Editor
Andrzej Zbrzezny
2018
Scientiﬁc Board
•Jiří Cihlář –Applications of Mathematics and Statistics, Education
in Mathematics – J. E. Purkyné University in Usti nad Labem,
Czech Republic
•Roman Frič –Probability Theory, Fuzzy Mathematics, Quantum
Structures – Slovak Academy of Science, Košice & Catholic Univer
sity in Ružomberok, Slovak Republic
•Joanna Grygiel –Universal Algebra, Logic – Jan Długosz Univer
sity in Częstochowa, Poland
•Ján Gunčaga –Mathematics Education, Teaching Methods, Peda
gogic Theory – Katolícka Univerzita v Ružomberku, Slovakia
•Jacek M. Jędrzejewski –Theory of Functions, Mathematical
Analysis – Jan Długosz University in Częstochowa, Poland
•Bohumil Novák –Education in Mathematics – Palacký University
in Olomouc, Czech Republic
•Yuriy Povstenko –Application of Mathematics – Jan Długosz
University in Częstochowa, Poland
•Grażyna Rygał –Education in Mathematics – Jan Długosz Uni
versity in Częstochowa, Poland
•Marcin Szpyrka –Computer Science – AGH University of Science
and Technology, Poland
•Oleg Tikhonenko –Computer Science, Applications of Mathema
tics – Częstochowa University of Technology, Poland
•Pavel Tlustý –Probability Theory, Optimization, Education in
Mathematics – University of South Bohemia in České Budějovice,
Czech Republic
•Marián Trenkler –Education in Mathematics, Graph Theory –
Catholic University in Ružomberok, Slovak Republic
•Bożena WoźnaSzcześniak –Computer Science – Jan Długosz
University in Częstochowa, Poland
•Małgorzata Wróbel –Mathematical Analysis – Częstochowa Uni
versity of Technology, Poland
•Tomáš Zdráhal –Algebra, Education in Mathematics – Palacký
University in Olomouc, Czech Republic
•Mahdi Zargayouna –Computer Science, Multiagent Systems –
Université ParisEst, France
•Andrzej Zbrzezny –Computer Science – Jan Długosz University
in Częstochowa, Poland
•Marcin Ziółkowski –Statistics, Computer Science – Jan Długosz
University in Częstochowa, Poland
3
Secretary: Katarzyna Domańska
Computer Typesetting and Technical Editing: Andrzej Zbrzezny
Graphical Project of a Cover: Sławomir Sadowski
ISSN 2450–9302
c
Copyright by Jan Długosz University in Częstochowa, 2018
Printed version is the primary one.
Cooperating Reviewers
Piotr Artiemjew — University of Warmia and Mazury in Olsztyn
Grzegorz Bryll — University of Opole
Yevhen Chaplia — National Academy of Sciences of Ukraine, Lviv, Ukraine
Imed El Fray — Warsaw University of Life Sciences
Roman Frič — Slovak Academy of Science, Košice;
& Catholic University in Ružomberok, Slovak Republic
Roman Ger — Silesian University in Katowice
Katarzyna Grygiel — Jagiellonian University
Ján Gunčaga — Catholic University in Ružomberok, Slowak Republic
Jacek Jędrzejewski — Jan Długosz University in Częstochowa (retired)
Magdalena Kacprzak — Bialystok University of Technology
Zygfryd Kominek — Silesian University in Katowice
Mirosław Kurkowski — Cardinal Stefan Wyszyński University
Janusz Matkowski — University of Zielona Góra
Tomasz Połacik — University of Silesia, Katowice
Tadeusz Poreda — Łódź University of Technology
Yuriy Povstenko — Jan Długosz University in Częstochowa
Yaroslav P’yanylo — Ukrainian National Academy of Sciences
Paweł Róg — Częstochowa University of Technology
Grażyna Rygał — Jan Długosz University in Częstochowa
Robert Sochacki — University of Opole
Marcin Szpyrka — AGH University of Science and Technology
Oleg Tikhonenko — Cardinal Stefan Wyszyński University in Warsaw
Szymon Wąsowicz — The University of BielskoBiała
Władysław Wilczyński — University of Łódź
Wacław Zawadowski — Warsaw University
Contents
Part I – Mathematics and Its Applications
1. J. M. Jędrzejewski –Asymmetry in Real Functions Theory 11
2. J. Jureczko –Some remarks on strong sequences 25
3. M. Liana, A. SzynalLiana, I. Włoch –On F(p, n)Fibonacci
bicomplex numbers 35
4. K. Pjanić, M. Vuković –Sangaku fan shape problems 45
5. K. TroczkaPawelec –Some remarks about Kcontinuity of K
superquadratic multifunctions 57
Part II – Computer Science
1. L. Stępień, M. R. Stępień –Automatic search of rational self
equivalences 67
Part I
Mathematics and its Application
Scientific Issues
Jan Długosz University
in Częstochowa
Mathematics XXIII (2018)
11–24
DOI http://dx.doi.org/10.16926/m.2018.23.01
ASYMMETRY IN REAL FUNCTIONS THEORY
JACEK MAREK JĘDRZEJEWSKI
Abstract
Since the beginning of the XX century many authors considered characterizations of
local properties for real functions of a real variable which have been deﬁned as global
properties. We present a short survey of local properties of the well known global ones
and consider of how small/big the set of asymmetrical behaviour of a function must be.
1. Introduction
We shall consider only real functions deﬁned in an open interval. When
we use topological terminology, then it is applied in the sense of natural
topology in the set of real numbers (or in its subsets).
Limit numbers of a real function deﬁned in subsets of Rhave been con
sidered in many articles by many mathematicians. Starting from the classi
cal result of W. H. Young [20] concerning asymmetry of functions through
problems of usual limit numbers, J. M. Jędrzejewski and W. Wilczyński
[12], approximate limit numbers discussed by M. Kulbacka [14], L. Belowska
[1], W. Wilczyński [18] and others, problems of qualitative limit numbers
(W. Wilczyński [19]) Blimit numbers (J. M. Jędrzejewski [7], [8], J. M.
Jędrzejewski together with W. Wilczyński [13]) one can come up to a big
monograph on local systems by B. S. Thomson [17].
The ﬁrst part of our considerations deals with the asymmetry of functions
with respect to limit numbers of diﬀerent kinds.
Some properties of functions (continuity, Darboux condition and others)
can be characterized globally and locally. For many of those properties we
have theorems which say that a function has this global property if and only
if it has its adequate local property. The second part of the article deals
with some of such properties.
•Jacek M. Jędrzejewski — jacek.m.jedrzejewski@gmail.com
Retired from Pomeranian Academy in Słupsk.
12 J. M. JĘDRZEJEWSKI
The last part of the paper is devoted to results obtained by T. Świątkowski
in view of general approach to limit numbers considered originally by B. S.
Thomson and me.
2. Asymmetry of Sets of Limit Numbers
2.1. Limit Numbers of a Real Function. We shall start with the clas
sical problem called Rome’s Theorem. The theorem was probably the ﬁrst
one which dealt with arbitrary function. Let us remind necessary deﬁnitions
and properties.
Deﬁnition 1. (W. H. Young [20]) Let a real function fbe deﬁned in an
open interval (a, b). Then a number g(or +∞or −∞) is called the limit
number of fat a point x0from (a, b)if there exists a sequence (tn)∞
n=1 such
that
(1) tn6=x0, for each positive integer n,
(2) limn→∞ tn=x0,
(3) limn→∞ f(tn) = g.
If the inequality tn6=x0is replaced by tn> x0, then such a limit number
is called the right limit number of fat x0.
If the inequality tn6=x0is replaced by tn< x0, then such a limit number
is called the left limit number of fat x0.
•By L+(f, x0)we denote the set of all right limit numbers of fat x0.
•By L−(f, x0)we denote the set of all left limit numbers of fat x0.
•By L(f, x0)we denote the set of all limit numbers of fat x0.
Let us remark that limit numbers can be equivalently deﬁned in the
following way:
Theorem 1. Let a real function fbe deﬁned in an open interval (a, b).
Then a number g(or +∞or −∞) is a limit number of fat a point x0from
(a, b)if and only if the set
x∈(a, b) : f−1(Ug)∩[(x0−ε, x0+ε)\ {x0}]
is nonempty for each positive εand each neighbourhood Ugof the point g.
It is obvious that:
Theorem 2. The sets L−(f, x0),L+(f , x0)and L(f, x0)are nonempty and
closed, moreover
L(f, x0) = L−(f , x0)∪L+(f, x0)
for any function f: (a, b)−→ Rand any x∈(a, b).
The main theorem which was announced in Rome at the congress of
mathematicians is stated as follows:
ASYMMETRY IN REAL FUNCTIONS THEORY 13
Theorem 3. Rome’s Theorem on Asymmetry (W. H. Young, 1906) For
any function f: (a, b)−→ Rthe set
x∈(a, b) : L−(f, x)6=L+(f , x)
is at most countable.
Quite similarly one can say that:
Theorem 4. For any function f: (a, b)−→ Rthe set
{x∈(a, b) : f(x0)/∈L(f, x)}
is at most countable.
Let us remark that for each countable set Ein Rthere exists a function
f:R−→ Rfor which
E=x∈(a, b) : L−(f, x)6=L+(f , x).
It is quite obvious if the set Eis ﬁnite; if it is inﬁnite it is possible to
deﬁne a monotone function, which fulﬁls the required condition. We shall
construct such a function.
Example 1. Monotone function with inﬁnite set of asymmetry.
Let E= (xn)∞
n=1 and the sequence of positive numbers (αn)∞
n=1 be such
that the series
∞
X
n=1
αnis convergent. The function
f(x) = X
{n:xn<x}
αn
fulﬁls all the required properties.
2.2. Qualitative Limit Numbers. Following the way as in Theorem 1.
one can deﬁne other kinds of limit numbers as qualitative (W. Wilczyński
[19]) or approximative limit numbers (L. Belowska [1], M. Kulbacka [14],
J. Jaskuła [5] and W. Wilczyński [18]) when we deﬁne limit numbers using
the above mentioned property.
Deﬁnition 2. A number gor +∞or −∞ is called the qualitative limit
number of a function fat a point x0if the set
x∈(a, b) : f−1(Ug)∩(x0−ε, x0+ε)
is of the second category for each positive εand arbitrary neighbourhood Ug
of the point g.
14 J. M. JĘDRZEJEWSKI
Deﬁnition 3. If the set
x∈(a, b) : f−1(U(g)) ∩(x0−ε, x0)
is of the second second category for each positive ε, then gis called the left
qualitative limit number of a function fat the point x0.
Similarly, gis called the right qualitative limit number of a function fat
a point x0if the set
x∈(a, b) : f−1(U(g)) ∩(x0, x0+ε)
is of the second category for each positive εand each neighbourhood Ugof
the point g.
•By L+
q(f, x0)we denote the set of all right qualitative limit numbers
of fat x0.
•By L−
q(f, x0)we denote the set of all left qualitative limit numbers
of fat x0.
•By Lq(f, x0)we denote the set of all qualitative limit numbers of f
at x0.
Then, similarly as for usual limit numbers one can state:
Theorem 5. For arbitrary real function fon the interval (a, b)and any
x0from (a, b)the sets Lq(f, x0),L−
q(f, x0)and L+
q(f, x0)are nonempty,
closed and
Lq(f, x0) = L−
q(f, x0)∪L+
q(f, x0).
Considering the sets of qualitative limit numbers we can get the analogue
of Rome’s Theorem, namely:
Theorem 6. For any function f: (a, b)−→ Rthe set
x∈(a, b) : L−
q(f, x)6=L+
q(f, x)
is at most countable.
We can observe that the considered sets are at most countable, it means
that they are rather small with natural topology in the set of real numbers.
The quantity of such sets will be of our main interest. Unfortunately not
always such sets must be countable.
2.3. Approximate Limit Numbers. Several mathematicians considered
approximate limit numbers but we remind basic deﬁnitions and properties.
Deﬁnition 4. A number gor +∞or −∞ is called the approximate limit
number of a function fat a point x0if the set
x∈(a, b) : f−1(U(g)) ∩[(x0−ε, x0+ε)]
ASYMMETRY IN REAL FUNCTIONS THEORY 15
has positive upper exterior density at x0for every open neighbourhood Ug
of the point gand each positive ε.
Deﬁnition 5. A number gor +∞or −∞ is called the left approximate
limit number of a function fat a point x0if the set
x∈(a, b) : f−1(U(g)) ∩[(x0−ε, x0)]
has positive upper exterior density at x0for every open neighbourhood Ug
of the point gand each positive ε.
And similarly, a number g(or +∞,−∞) is called the right approximate
limit number of a function fat a point x0if the set
x∈(a, b) : f−1(U(g)) ∩[(x0, x0+ε)]
has positive upper exterior density at x0for every open neighbourhood Ug
of the point gand each positive ε.
•By L+
a(f, x0)we denote the set of all right approximate limit num
bers of fat x0.
•By L−
a(f, x0)we denote the set of all left approximate limit numbers
of fat x0.
•By La(f, x0)we denote the set of all approximate limit numbers of
fat x0.
Then, similarly as for usual limit numbers one can state:
Theorem 7. For arbitrary real function fon the interval (a, b)and any
x0from (a, b)the sets La(f, x0),L−
a(f, x0)and L+
a(f, x0)are nonempty,
closed and
La(f, x0) = L−
a(f, x0)∪L+
a(f, x0).
Now considering the sets of approximate limit numbers we can get the
analogue of Rome’s Theorem, but:
Theorem 8. (M. Kulbacka [14]). For any function f: (a, b)−→ Rthe set
x∈(a, b) : L−
a(f, x)6=L+
a(f, x)
is ﬁrst category set and has measure 0.
This time the sets of the ﬁrst category which have measure 0do not char
acterize the set of asymmetry of functions. J. Jaskuła gave some additional
properties for the set of approximate asymmetry.
Theorem 9. (J. Jaskuła [5]) For any function f: (a, b)−→ Rthe set
x∈(a, b) : L−
a(f, x)6=L+
a(f, x)
is ﬁrst category and has measure 0, moreover it is of type Fσδσ .1
1W.Wilczyński informed me that the results of J. Jaskuła were a big deeper, i.e. the
set approximate asymmetry is also σporous.
16 J. M. JĘDRZEJEWSKI
2.4. Generalized Limit Numbers. Let us observe that the class of sets
which are of the ﬁrst category at the point x0and the class of positive upper
external density at that point have common properties. When we denote
such a class by Bthen this class fulﬁls:
(1) If B∈ B and E⊃B, then E∈ B,
(2) If B1∪B2∈ B then B1∈ B or B2∈ B,
(3) If B∈ B and ε > 0then B∩(x0−ε, x0+ε)∈ B.
The class of sets which are uncountable in each (x0−ε, x0+ε)or have
positive outer measure in each such interval and many other classes of sets
have the previously pointed properties. The articles on this topic are as
follows: J. Jędrzejewski [7], [8], J. Jędrzejewski with W. Wilczyński [13], J.
Jędrzejewski with S. Kowalczyk [10] and [11].
Let us start now from the beginning:
Deﬁnition 6. For each x∈Rlet B+
xbe a class of nonempty sets fulﬁlling
the following conditions:
(1) B1∪B2∈B+
x⇐⇒ (B1∈B+
x∨B2∈B+
x),
(2) B∩(x, x +t)∈B+
xfor each B∈B+
xand t > 0.
For each x∈Rlet B−
xbe a class of nonempty sets fulﬁlling the following
conditions:
(1) B1∪B2∈B−
x⇐⇒ (B1∈B−
x∨B2∈B−
x),
(2) B∩(x, x +t)∈B−
xfor each B∈B−
xand t > 0.
Let Bx=B−
x∪B+
x.
Deﬁnition 7. If fdeﬁned in some (a, b)is a real function, then a number
(or +∞or −∞) is called Blimit number of fat x0from (a, b)if
x∈(a, b) : f−1(Ug)∈Bx0
for any neighbourhood Ugof the point g.
Deﬁnition 8. If
x∈(a, b) : f−1(Ug)∈B−
x0
for any neighbourhood Ugof the point g, then gis called the left Blimit
number of a function fat a point x0.
Similarly we deﬁne right Blimit numbers of a function fat a point x0.
•By L+
B(f, x0)we denote the set of all right Blimit numbers of fat
x0.
•By L−
B(f, x0)we denote the set of all left Blimit numbers of fat
x0.
•By LB(f, x0)we denote the set of all Blimit numbers of fat x0.
Then, as for usual limit numbers, one can state:
ASYMMETRY IN REAL FUNCTIONS THEORY 17
Theorem 10. For arbitrary real function fon the interval (a, b)and any
x0from (a, b)the sets LB(f, x0),L−
B(f, x0)and L+
B(f, x0)are nonempty,
closed and
LB(f, x0) = L−
B(f, x0)∪L+
B(f, x0).
Considering the sets of Blimit numbers we are not able to get the ana
logue of Rome’s Theorem. The situation depends on the class B. But if we
add a special condition for the family B, we can get adequate analogue of
Young’s theorem.
Deﬁnition 9. We say that the class Bfulﬁls condition Mif
∞
[
n=1
En∈Bx0
for any: x0∈(a, b), sequence (xn)∞
n=1 converging to x0and every sequence
of sets (En)∞
n=1 such that En∈Bxn.
This condition permits us to state:
Theorem 11. If the class Bfulﬁls condition M, then
x∈(a, b) : L−
B(f, x)6=L+
B(f, x)
is at most countable set for any function f: (a, b)−→ R.
3. Asymmetry for Some Classes of Functions
3.1. Diﬀerentiation of Functions. Everybody knows:
Theorem 12. The set of all those points at which left derivative of a func
tion f:R−→ Ris diﬀerent from the right derivative of this function is at
most countable.
3.2. Continuity of Functions. One can get that the set of points at which
a function is continuous from exactly one side as a quite simple corollary of
Young’s Theorem.
Theorem 13. For any function f:R−→ Rthe set of all points at which
fis continuous from the only one side is at most countable.
3.3. Darboux Condition of Functions. As before: everybody knows
that Darboux condition has been originally deﬁned as a global condition of
a function. It sounds like this: the function ffulﬁls Darboux condition if
it takes all values in between; exactly:
Deﬁnition 10. We say that a function f: (a, b)−→ Rfulﬁls Darboux con
dition if for any x1and x2such that f(x1)6=f(x2)and any number clying
between f(x1)and f(x2)there exists a point xlying (strictly) between x1
and x2such that f(x) = c.
18 J. M. JĘDRZEJEWSKI
This condition can be replaced by the one that function ftransforms
connected sets onto connected sets.
But still this condition is not good enough to say about asymmetry. We
should deﬁne this condition locally, even more it must be deﬁned separately
for both sides. Let’s start to do it, what was done by A. Bruckner and J.
Ceder in 1965. For simplicity, let us assume that all the discussed functions
are bounded.
Deﬁnition 11. (A. Bruckner, J. Ceder) [2]) A function f: (a, b)−→ Ris
said to be Darboux from the left side at a point x0∈(a, b)if
(1) f(x0)∈L−(f, x0),
(2) for each c∈(inf L−(f, x0),sup L−(f, x0)) and for each t > 0there
exists a point x∈(x0−t, x0)such that f(x) = c.
Similarly,
Deﬁnition 12. We say that a function f: (a, b)−→ Ris Darboux from the
right side at a point x0∈(a, b)if
(1) f(x0)∈L+(f, x0),
(2) for each c∈(inf L+(f, x0),sup L+(f, x0)) and for each t > 0there
is a point x∈(x0, x0+t)such that f(x) = c.
In the end:
Deﬁnition 13. We say that a function f: (a, b)−→ Ris Darboux at a point
x0∈(a, b)if it is Darboux from both sides at x0.
These deﬁnitions would not be good enough if the next theorem is false.
But luckily it is not so.
Theorem 14. A function f: (a, b)−→ Ris Darboux if and only if it is
Darboux at each point x0∈(a, b).
And now we can say about Darboux asymmetry.
Theorem 15. [9]. For each function f: (a, b)−→ Rthe set of all those
points at which fDarboux from exactly one side is at most countable.
3.4. Connectedness of Functions. Next class of functions we want to
discuss is the class of functions with connected graphs. They are called
connected functions, however they can be deﬁned in each topological spaces
we shall consider only real functions deﬁned in an interval. The adequate
characterization has been given by B. D. Garret, D. Nelms and K. R. Kel
lum [3].
Deﬁnition 14. A function f: (a, b)−→ Ris called connected if its graph
is a connected set on the plane.
ASYMMETRY IN REAL FUNCTIONS THEORY 19
As before this deﬁnition is a global one, we have to ﬁnd a local deﬁnition
which will be as good as to get that local and global characterizations
coincide.
As before, we assume that all discussed functions are bounded.
Deﬁnition 15. (B. D. Garret, D. Nelms, K. R. Kellum) [3]) A function
f: (a, b)−→ Ris connected from the left side at a point x0∈(a, b)if
(1) f(x0)∈L−(f, x0),
(2) for each continuum K(connected and compact set) such that
projx(K) = [x0−t, x0] for some t > 0
and
projy(K)⊂inf L−(f , x0),sup L−(f, x0)
the (graph) function fhas common point with K.
Similarly:
Deﬁnition 16. A function f: (a, b)−→ Ris connected from the right side
at a point x0∈(a, b)if
(1) f(x0)∈L+(f, x0),
(2) for each continuum Ksuch that
projx(K) = [x0, x0+t] for some t > 0
and
projy(K)⊂inf L+(f , x0),sup L+(f, x0)
the (graph) function fhas common point with K.
Deﬁnition 17. We say that a function f: (a, b)−→ Ris connected at
a point x0∈(a, b)if it is connected from both sides at x0.
And of course:
Theorem 16. A function f: (a, b)−→ Ris connected if and only if it is
connected at each point x0∈(a, b).
Finally, we are able to formulate theorem on connectivity asymmetry.
Theorem 17. For each function f: (a, b)−→ Rthe set of all those points
at which fis connected from exactly one side is at most countable.
3.5. Almost Continuity of Functions. The last class of functions we
want to discuss is the class of almost continuous functions. The adequate
local characterization has been given by J. M. Jastrzębski, T. Natkaniec
and J. Jędrzejewski [6].
20 J. M. JĘDRZEJEWSKI
Deﬁnition 18. A function f: (a, b)−→ Ris called almost continuous if
each neighbourhood of its graph contains some continuous function deﬁned
in (a, b).
As before this deﬁnition is a global one, we have to ﬁnd a local deﬁnition
which will be as good as to get that local and global characterizations
coincide.
We assume that all discussed functions are bounded.
Deﬁnition 19. A function f: (a, b)−→ Ris almost continuous from the
left side at a point x0∈(a, b)if
(1) f(x0)∈L−(f, x0),
(2) there is a positive εsuch that for each open neighbourhood of f(x,∞)
arbitrary y∈(inf L−(f, x0),sup L−(f, x0)), arbitrary neighbourhood
Gof the point (x, y)∈R2and arbitrary t∈(x0, x0+ε)there is a
continuous function g: (x0, x0+ε)−→ Rsuch that g⊂U∪Gand
g(x0) = y,g(t) = f(t).
Similarly one can deﬁne almost continuity from the right side at a point
x0∈(a, b).
Deﬁnition 20. We say that a function f: (a, b)−→ Ris almost continuous
at a point x0∈(a, b)if it is almost continuous from both sides at x0.
And of course:
Theorem 18. A function f: (a, b)−→ Ris almost continuous if and only
if it is almost continuous at each point x0∈(a, b).
Finally, one can state:
Theorem 19. For each function f: (a, b)−→ Rthe set of all those points
at which fis almost continuous from exactly one side is at most countable.
4. General Approach to Asymmetry of Functions
Some general theorems were discussed in previous parts of the article.
Let us come to Thomson’s monograph. B. S. Thomson gathered several
ideas in one theory. He deﬁned local systems which contain Bclasses and
B∗classes that have been deﬁned in [7]. For sake of completeness let us
remind the basic notions.
4.1. Local Systems.
Deﬁnition 21. B. S. Thomson [17].
By a local system in Rwe mean a class Sconsisting of nonempty collections
S(x)for each real number x, fulﬁlling the following conditions:
(1) {x}/∈ S(x),
ASYMMETRY IN REAL FUNCTIONS THEORY 21
(2) E∈ S(x) =⇒x∈E,
(3) (E∈ S(x)∧F⊃E) =⇒F∈ S(x),
(4) (E∈ S(x)∧δ > 0) =⇒E∩(x−δ, x +δ)∈ S(x).
Deﬁnition 22. By a left local system in Rwe mean a class Sconsisting of
nonempty collections S(x)for each real number x, fulﬁlling the following
conditions:
(5) {x}/∈ S(x),
(6) E∈ S(x) =⇒x∈E,
(7) (E∈ S(x)∧F⊃E) =⇒F∈ S(x),
(8) (E∈ S(x)∧δ > 0) =⇒E∩(x−δ, x]∈ S(x).
Similarly we deﬁne right local systems.
A local system is called ﬁltering at a point xif
(9) E∩F∈ S(x)whenever E ∈ S(x)and F ∈ S (x).
A local system is called ﬁltering if it is ﬁltering at each xin R.
A local system is called bilateral if
E∩(x−δ, x)6=∅and E ∩(x, x +δ)6=∅
for each x∈R, E ∈ S(x)and δ > 0.
Let us observe that those deﬁnitions are very close to Deﬁnition 6. When
B. S. Thomson assumes that dual system for Sis ﬁltering, then Sfulﬁls all
requirements of Deﬁnition 6. The only diﬀerence lays in the belonging of
the point xto every set from the class Sx.
Deﬁnition 23. A number gis called Slimit of a function fat a point xif
f−1(g−ε, g +ε)∪ {x} ∈ S(x)
for each positive ε.
We shall write then
g= (S) lim
t→xf(t).
The set of all (S)limits are denoted by ΛS(f, x).
For each local system Sthere is a system S∗which is also a local system,
that is deﬁned by:
E∈ S∗(x)⇐⇒ x∈E∧(R\E)∪ {x}/∈ S(x).
This system is called dual system for S.
22 J. M. JĘDRZEJEWSKI
A system Sis called ﬁltering if E1∩E2∈ S(x)for every sets E1∈ S(x)
and E2∈ S(x)and each x∈R.
Deﬁnition 24. We say that two systems S1and S2satisfy a joint inter
section condition if for any choices {Ex:x∈R}and {Dx:x∈R}such
that Ex∈ S1(x),Dx∈ S2(x)there exists a gauge δon Rso that if
0<x−y<min{δ(x), δ(y)}then at least one of the sets Ex∩Dyor
Dx∩Eycontains points other than xand y.
By a gauge on the set Rwe mean a positive function deﬁned in R.
And now we are able to formulate the asymmetry theorem given by
Thomson.
Theorem 20. Let S1,S2be local systems such that both of them are ﬁltering
and that the pair (S1,S2)has the joint intersection condition. Then for any
function f:R−→ Rthe set
{x∈R: ΛS1(f, x)6= ΛS2(f , x)}
is at most countable.
Example 2.
Let S1
xbe the class consisting of all sets Efor which E∩(x−ε, x +ε)is
of the ﬁrst category.
Let S2
xbe the class consisting of all sets Dfor which D∩(x−ε, x +ε)
has positive outer measure.
There are two sets Aand Bsuch that A∩B=∅,A∪B= (0,1),Ais
of the ﬁrst category in (0,1), and Bhas measure 1.
Let f: (0,1) −→ Rbe deﬁned as follows:
f(x) = 0 if x∈A,
1 if x∈B.
For this function, all points from (0,1) are points of S1,S2asymmetry.
4.2. Świątkowski Approach to Asymmetry.
Deﬁnition 25. (T. Świątkowski [15]) Let Tbe a stronger topology in Rthan
the natural one. For a subset Eof Rthe symbol E0
Tdenote the set of all
accumulation points with respect to topology T. Let moreover Lx= (−∞, x)
and Px= (x, ∞)for any real number x. Consider now the function ϕin
the following way:
x∈ϕ(A)if x ∈(A∩Lx)0
T4(A∩Px)0
T
for any subset Aof R.
Each point from the set (A∩Lx)0
T4(A∩Px)0
Tis called Tasymmetry
point of the set A.
ASYMMETRY IN REAL FUNCTIONS THEORY 23
Deﬁnition 26. Let f:R−→ Rbe arbitrary function and xa real number.
We say that gis Tlimit number of the function fat a point xif
x∈f−1(U)0
T
for each neighbourhood Uof the point x.
Not every topology is good enough to get the adequate theorem on asym
metry; let us call the property (W)from the article [15].
Deﬁnition 27. [15] Let Tbe a stronger topology then the natural one in
the set R. We say that Tfulﬁls condition (W)if for every x∈R, sequence
(xn)∞
n=1 converging to xand every sequence (En)∞
n=1 such that xn∈(En)0
T
the point xbelongs to (S∞
n=1 En)0
T.
This condition (W)for the topology Tdescribed as above is equivalent
to the condition (W0):
for an arbitrary x∈Rand its Tneighbourhood Uthere exists a positive
number δsuch that ((x−δ, x +δ)\U)0
T=∅.
The condition (W0)allows to formulate one of the most general theorems
on asymmetry.
Theorem 21. If Tis a stronger than the natural topology in the set Rand
fulﬁls condition (W), then for any function f:R−→ Rthe set of asymmetry
of fis at most countable.
It is now easy to observe that:
If Tis a natural topology in R, Theorem 21 allows us to obtain the
classical Young’s Theorem on asymmetry. It is implied from the fact that
Tfulﬁls condition (W0)(see Theorem 3).
Let us remark that if Tis a Hashimoto topology in Rgenerated by sets of
the ﬁrst category, Theorem 21 allows us to obtain Theorem on qualitative
asymmetry of functions. It follows from the fact that Talso fulﬁls (W0)
(see Theorem 6).
4.3. Comments on the Three Approaches to Asymmetry. When
we want to compare the three ideas of B. S. Thomson, of T. Świątkowski
and J. Jędrzejewski, we can observe that some local systems Slimits can
be understood as Blimits, some systems can be understood as systems B.
However, in each theorem where Thomson assumes that the dual system for
a system Sis ﬁltering, then the system fulﬁls all conditions for the system
B. Świątkowski’s condition and mine called Wor Mare equivalent, so
Thomson’s theorems are almost the same as Świątkowski’s and mine ones.
The only diﬀerence lays on diﬀerent approaches to the problem.
24 J. M. JĘDRZEJEWSKI
References
[1] Belowska L., Résolution d’un problème de M. Z. Zahorski sur les limites
approximatives, Fund. Math. 48 (1960), p. 277 – 286,
[2] Bruckner A., Ceder J., Darboux continuity, Jber. Deutsch. Math. Verein. 67
(1965), p. 93 – 117,
[3] Garret B. D., Nelms D., Kellum K. R., Characterizations of connected real
functions Jber. Deutsch. Math. Verein. 79 (1971), p. 131 – 137,
[4] Hashimoto H., On the *topology and application, Fund. Math. XCI, (1976),
p. 5–10,
[5] Jaskuła J., Doctor’s Thesis, Uniwersytet Łódzki, 1971,
[6] Jastrzębski J. M., Jędrzejewski J. M., Natkaniec T., Points of Almost Con
tinuity of Real Functions, Real Anal. Ex. 16 (1990–91),
[7] Jędrzejewski J. M., On limit numbers of real functions, Fund. Math.
LXXXIII (1974), p. 269 – 281,
[8] Jędrzejewski J. M., On the family of sets of Blimit values and Baire’s
functions, Acta Univ. Lodz. ser. II 14 (1980), p. 59 – 66,
[9] Jędrzejewski J. M., On Darboux Asymmetry, Real Anal. Ex. 7 (198182),
172 – 176,
[10] Jędrzejewski J. M., Kowalczyk S., Generalized Cluster Sets of Real Func
tions, Tatra Mount. Math. Publ. 62 (2014), p. 1 – 7, DOI: 10.2478/tmmp
20140000.
[11] Jędrzejewski J. M., Kowalczyk S., Cluster Sets and Topology, Mathematica
Slovaca, in print
[12] Jędrzejewski J., Wilczyński W., On the family of sets of limit numbers, Bull.
Acad. Polon. Sci. sér. Sci. Math. Astronom. Phys. 18 (8) (1970), p. 453 –
460,
[13] Jędrzejewski J. M., Wilczyński W., On the family of sets of Blimit numbers,
Zeszyty Nauk. Uniw. Łódzkiego 52 (1973),
[14] Kulbacka M., Sur l’ensemble des points de l’asymétrie approximative, Acta
Sci. Math. (Szeged) 21 (1960), p. 90 – 95,
[15] Świątkowski T., On some generalization of the notion of asymmetry of func
tions, Coll. Math. 17 (1967), p. 77 – 91,
[16] Świątkowski T., On a certain generalization of the notion of derivative,
Zeszyty Naukowe PŁ, 149, Mat. (1972), 89 – 103,
[17] Thomson B. S., Real Functions, Lectures Notes in Mathematics, No 1170,
Springer Verlag, BerlinHeidelbergNew YorkTokyo 1985,
[18] Wilczyński W., On the Family of Sets of Qualitative Limit Numbers, Rev.
Roum. de Math. Pures et App., XVIII, (1973), p. 1297 – 1302,
[19] Wilczyński W., On the Family of Sets of Approximate Limit Numbers, Fund.
Math. LXXV (1972), p. 169 – 174,
[20] Young W. H., La symétrie de structure des fonctions des variables réelles
Bull. Sci. Math. 52 (2) (1928), p. 265 – 280,
Received: July 2018
Email address:jacek.m.jedrzejewski@gmail.com
Scientific Issues
Jan Długosz University
in Częstochowa
Mathematics XXIII (2018)
25–34
DOI http://dx.doi.org/10.16926/m.2018.23.02
SOME REMARKS ON STRONG SEQUENCES
JOANNA JURECZKO
Abstract
Strong sequences were introduced by Eﬁmov in the 60s’ of the last century as a useful
method for proving well known theorems on dyadic spaces i.e. continuous images of the
Cantor cube. The aim of this paper is to show relations between the cardinal invariant
associated with strong sequences and well known invariants of the continuum.
1. Introduction
Strong sequences were introduced by B. A. Eﬁmov in [4], as a useful
tool for proving well known theorems on dyadic spaces. Among others he
proved that strong sequences do not exist in the subbase of the Cantor
cube. This is our opinion that it could be interesting the answer of the
natural question about properties of spaces in which strong sequences exist
and consequences of such existence. This is how the interest of the strong
sequences method was born, (for further historical notes concerning strong
sequences see [6]). Particularly, strong sequences method, as was shown in
e.g. [7, 8] is equivalent to partition theorems. Moreover, if we associate the
cardinal invariant with the length of strong sequences in spaces where such
sequences exist, we can obtain interesting results, (see also [8, 9]). This is
our hope that this invariant can be usefull characterisation of such spaces.
In this paper we will consider the space (ωω,≤∗)in which, as we will
show, strong sequences exist. We will investigate inequalities between in
variant ˆs associated with strong sequences and other well known invariants
like: boundeness, covering number and the invariant associated with MAD
families.
Our paper is organized as follows. In section 2 we gather all deﬁnitions
and previous facts needed for further parts of this paper. In Section 3
we show main results. The paper is ﬁnished by some results in forcing,
•Joanna Jureczko — email: joanna.jureczko@pwr.edu.pl
Wrocław University of Science and Technology.
26 J. JURECZKO
(Section 4) in which we will show some strong inequalities which can be
obtained between ˆs and considered invariants. In this part we give some
open problems.
2. Definitions and previous results
1. Consider a partially preordered set (X, ), i.e. a set ordered by reﬂex
ive and transitive relation . Let a, b, c, x ∈X. We say that aand bare
comparable iﬀ abor ba. We say that aand bare compatible iﬀ there
exists c∈Xsuch that acand bc. (In this case we say that a, b have
abound). A set A⊂Xis called an ωdirected set iﬀ every subset of Aof
cardinality less than ωhas a bound which belongs to A.
Deﬁnition 1. A sequence (Sφ, Hφ)φ<α, where Sφ, Hφ⊂X, and Sφ< ω
is called a strong sequence if:
1oSφ∪Hφis ωdirected for all φ<α;
2oSψ∪Hφis not ωdirected, for all ψand φsuch that φ<ψ<α.
In [6] the strong sequence number ˆs(X)was introduced as follows:
(1) ˆs(X) = sup {κ:there exists a strong sequence on Xof length κ}.
2. We say that (X, )iﬀ is reﬂexive and transitive.
A subset B⊂Xis called bounded iﬀ Bhas a bound. The set which is not
bounded will be called unbounded.
A subset A⊆B⊆Xis called coﬁnal in Biﬀ for any b∈Bthere exists
a∈Asuch that ba. A coﬁnal subset in the whole set Xis called also a
dominating set. The following invariants are well known:
(2) b(X) = min {A:A⊂X∧Ais unbounded in X},
(3) d(X) = min {A:A⊂X∧Ais coﬁnal in X}.
Fact 1 ([3]). Let (X, )be a partially preordered set without maximal
elements. Then b(X)is regular and
(4) b(X)≤cf (d(X)) ≤d(X).
3. We will provide our considerations for (X, ) = (ωω,≤∗), i.e. in the
set of all functions ω→ωordered by
(5) f≤∗giﬀ  {n∈ω:g(n)< f (n)}  < ω.
We accept the notation: ˆs= ˆs(ωω),b=b(ωω),d=d(ωω).
4. A family Iof subsets of Xwhich satisﬁes the following three condi
tions
SOME REMARKS ON STRONG SEQUENCES 27
1) A∈ I and B⊂Athen B∈ I;
2) {x} ∈ I for all x∈ I;
3) X6∈ I
is called a family of thin sets.
A subfamily B ⊂ I is called a base of the family Iof thin sets iﬀ for each
set A∈ I there exists a set B∈ B such that A⊆B.
We remind deﬁnitions of the following invariants, (see e. g. [3] p.250):
(6) add (I) = min nA:A ⊂ I ∧ [A 6∈ Io
(7) cov (I) = min nA :A ⊂ I ∧ [A=Xo
(8) non (I) = min {A:A6∈ I ∧ A∈ P (X)}
(9) cof (I) = min {A:A ⊂ I ∧ A is a base of I } .
Notice that any ideal on Xis a family of thin sets. (Clearly, Iis an ideal
iﬀ add (I)≥ ℵ0).
The following diagram is known in the literature as "Cichoń diagram"
and was introduced by Fremlin in [5]. Since that paper the diagram has
been completed and modiﬁed by many authors. Below we remind this dia
gram for four invariants deﬁned above.
Fact 2 ([1]). If Iis a family of thin sets, then
ℵ0add(I)
cov(I)
non(I)
cof(I)2ℵ0

@@@
@R
@@@
@R
where α→βdenotes α≤β.
5. Let Rbe the real line with standard topology. Let µbe the Lebesque
measure on R. Then
(10) M={A⊂R:Ais meager},
(11) N={A⊂R:µ(A) = ∅} .
28 J. JURECZKO
Notice, that Mand Nare both ideals.
6. In [1] one can ﬁnd the following results:
Fact 3 (Bartoszyński) cov (M)is the cardinality of the smallest fam
ily F ⊆ ωωsuch that
(12) ∀g∈ωω∃f∈F  {n∈ω:f(n)6=g(n)}  < ω.
Fact 4 (Keremedis) non (M)is the cardinality of the smallest family
F ⊆ ωωsuch that
(13) ∀g∈ωω∃f∈F  {n∈ω:f(n) = g(n)}  < ω.
Fact 5 (Rothberger)
(14) cov (M)≤non (N)and cov (N)≤non (M).
Fact 6 (Bartoszyński, Raisonnier and Stern)
(15) add (N)≤add (M),
(16) cof (M)≤cof(N).
Fact 7 (Miller, Truss)
(17) add (M) = min {cov (M),b}.
Fact 8 (Fremlin)
(18) cof (M) = max {non (M),d}.
According to equalities (14)  (18) the following diagram holds:
Fact 9 ([1]).
cov(N)non(M)cof(M)cof(N)
add(N)add(M)cov(M)non(N)
bd
 
6
6
6
6
6

 
6

where α→βdenotes α≤β.
SOME REMARKS ON STRONG SEQUENCES 29
Observation 1. (i) Let F ⊆ ωωbe the smallest family of the property
∀g∈ωω∃f∈F  {n∈ω:f(n) = g(n)}  < ω.
Then  {n∈ω:fα(n)6=fβ(n)}  =ωfor all fα, fβ∈ F , α 6=β.
(ii) Let F ⊆ ωωbe the smallest family of the property
∀g∈ωω∃f∈F  {n∈ω:f(n)6=g(n)}  < ω.
Then  {n∈ω:fα(n)6=fβ(n)}  =ωfor all fα, fβ∈ F , α =β.
Proof. We prove (i)only, (ii)can be proved similarly but using Fact 3.
(i)By Fact 4 we have F =non (M). Suppose in contrary that there are
α6=βsuch that  {n∈ω:fα(n) = fβ(n)}  =ω. Let
A(α, β) = {n∈ω:fα(n) = fβ(n)}.
Let {gγ∈ωω\ F :γ < η}be a family such that
 {n∈ω:gγ(n) = fβ(n)}  < ω
for all γ < η. Let B(γ, β) = {n∈ω:gγ(n) = fβ(n)}for all γ < η.
Obviously A(α, β)∩B(γ, β)< ω. Then gγ(n) = fα(n)for all n∈
A(α, β)∩B(γ , β). A contradiction with the minimality of F.
7. Two functions f, g ∈ωωare almost disjoint iﬀ there are ﬁnite values
of α∈Dom (f)∩Dom (g)such that f(α) = g(α). When the functions
have domain ωalmost disjointness means that they are eventually diﬀerent
(f(α)6=g(α)) for all suﬃciently large α < ω. A maximal almost disjoint
(MAD) family of functions on ωis an almost disjoint family of functions
ω→ωthat is not properly included in another such family. In [2] the
following invariant is associated with MAD families of functions:
(19) ae= min {A ⊆ P(ωω) : Ais a MAD family}.
Fact 10 ([2]).
(20) ae≥ω+.
Observation 2.
(21) non (M)≤ae.
Proof. Immediately by Fact 4.
30 J. JURECZKO
3. Main results
Theorem 1.
(22) b≤ˆs.
Proof. Suppose that ˆs<band κ≤ˆs. Let {(Sα, Hα) : α < κ}be a maximal
strong sequence in ωω. For any α < κ deﬁne
Aα={f∈Sα\Hβ:{f} ∪ Hβis not ωdirected for β < α}.
Deﬁne an increasing function
F:κ→[
α<κ
(Sα∪Hα).
such that
F(α) = fα∈Hαfor α= 0;
fα∈Aαfor α > 0.
Since ωωhas no maximal elements, this function is welldeﬁned.
Let
A={fα∈Aα:fα=F(α), α < κ}.
Since κ < b, there exists g∈Asuch that fα≤g, for all fα∈Sα. As ωωhas
no maximal elements, there exists h∈ωω\Sα<κ (Sα∪Hα)such that g < h.
Thus there exists a maximal ωdirected set S⊂ωω\Sα<κ (Sα∪Hα)such
that h∈Sand S∪Hαis not ωdirected for any α < κ. A contradiction
with maximality of the strong sequence {(Sα, Hα) : α < κ}.
Theorem 2.
(23) cov (M)≤ˆs.
Proof. Let cov (M) = κ. By Fact 3 there exists the smallest family
F={fα∈ωω:α < κ}
fulﬁlling (12)
Thus we can construct a function H:ωω→κsuch that
H(g) = min {α: {n∈ω:fα(n) = g(n)}  =ω}.
The family Fis wellordered hence the function His welldeﬁned.
We will construct a strong sequence in ωωwith relation deﬁned as follows:
if fα∈ F,then fαgiﬀ h(g) = α;
if f6∈ F,then fgiﬀ  {n∈ω:f(n) = g(n)}  =ω.
Let g0∈ωωbe an arbitrary function. Then there exists f∈ F such
that  {n∈ω:f(n) = g0(n)}  =ω. Let fα0∈ F be a function such that
h(g0) = α0.Let S0={g0}and H0={g∈ωω:h(g) = α0}.Obviously
H0is nonempty. Let (S0,H0)be the ﬁrst element of a strong sequence.
SOME REMARKS ON STRONG SEQUENCES 31
Since H06=ωωthere exists g1∈ωω\ H0such that h(g1)6=α0. Hence
we can construct the next element of the strong sequence. Let fα1∈ F be
a fucntion such that  {n∈ω:g1(n) = fα1(n)}  =ω. Let S1={g1}and
H1={g∈ωω\ H0:h(g) = α1}.
Assume that the strong sequence {(Sγ,Hγ) : γ < β}such that
(Sγ,Hγ) = {gγ},ng∈ωω\[{Hδ:δ < γ}:h(g) = αγo,
where gγ∈ωω\Sδ<γ Hδ, has been deﬁned,.
Since β < κ and by Observation 1, there exists gβ∈ωω\fαγ:γ < β
be a function such that n∈ω:gβ(n) = fαβ(n)=ω. Let
(Sβ,Hβ) = {gβ},ng∈ωω\[{Hγ:γ < βo:h(g) = αβ}.
Thus the strong sequence of length F has been constructed.
Theorem 3.
(24) ae≤ˆs.
Proof. By Fact 8 we have ae≥ω+.Let Febe a MAD family of functions
ω→ωof cardinality ω+. We will construct a strong sequence of cardinality
ω+in ωωwith the following relatio:
fgiﬀ  {α∈ω:f(α) = g(α)}  =ω.
Let f0∈ Febe a function. Let (S0,H0)=({f0},{g∈ωω:f0g})be the
ﬁrst element of a strong sequence. Obviously (S0,H0)is nonempty because
f0∈ H0. Let f1∈ Fe\ H0. Let (S1,H1) = ({f1},{g∈ωω:f1g}).By
our construction H0∪ H1is not ωdirected. Let (S1,H1)be the second
element of the strong sequence.
Assume that the strong sequence {(Sγ,Hγ) : γ < β < ω+}such that
(Sγ,Hγ) = {fγ},ng∈ωω\[{Hδ:δ < γ :fγgo,
where fγ∈ Fe\S{Hδ:δ < γ}, has been deﬁned.
Since β < ω+there exists fβ∈ Fe\S{Hγ:γ < β}. Let
(Sβ,Hβ) = {fβ},ng∈ωω\[{Hδ:γ < β :fβgo,
Thus the strong sequence of length F has been constructed.
Corollary 1.
(25) non (M)≤ae≤ˆs.
Proof. Immediately by Fact 10 and Theorem 3.
Theorem 4. In (ωω,≤∗)there exists a strong sequence of length 2ℵ0.
32 J. JURECZKO
Proof. Fix a MAD family of sets A=Aα⊆[ω]ω:α < 2ℵ0, (i.e. a family
of inﬁnite subsets of ωsuch that A∩B< ω for any A, B ∈ A). For each
A∈ A consider functions: FA
n∈ωωsuch that
FA
n(a) = n+ 1 for a∈A
0for a6∈ A
and FA
ω∈ωωsuch that
FA
ω(a) = afor a∈A
0for a6∈ A.
Obviously
FA
0<∗FA
1<∗... <∗FA
ω.
Now take (SA, HA) = {FA
ω},FA
n:n<ω. Then SA∪HAis ωdirected,
because FA
ωis its bound. Now take Aα, Aβ∈ A such that α < β. Then
SAβ∪HAαis not ωdirected, because it contains no bound for HAα. Since
all MAD families have cardinality 2ℵ0we obtain that {(SA, HA) : A∈ A}
is the required strong sequence.
Corollary 2. The following diagram holds
non(M)cof(M)
add(M)cov(M)
bd
ˆs
ae
2ℵ0
ℵ0

6
6
6
6

 

@
@
@
@
@I
XXXXz XXXXz
:
where α→βmeans α≤β:
Proof. Immediately by equalities (4), (17), (18) and Theorems 14.
4. Some results for forcing notion
According to [1] pp. 380397, the following inequalities are consistent
with ZFC.
In the iterated Cohen’s model with ﬁnite supports non (M) = ℵ1∧
cov (M) = cwhich is connecting with Cichoń diagram we have add (N) =
add (M) = cov (M) = non (M) = b=ℵ1and cov (M) = r=cof (M) =
cof (N) = non (N) = c>ℵ1.Thus
(26) add (N) = add (M) = cov (N) = cov (M) = b<ˆs.
By adding ℵ2random reals a model of CH we have non(N) = ℵ1<
cov (N) = ℵ2=c. Thus
(27) non (N)<ˆs.
By adding ℵ2Hechler’s reals (with ﬁnite support) to a model of CH we
get cov (N) = ℵ1<add (M) = ℵ2=c. Hence it is consistent that
(28) cov (N)<ˆs.
Alternatively adding ℵ2Cohen and Laver reals (with countable support)
over a model of CH we have cov(N) = ℵ1<add (M) = ℵ2=cThus
(29) cov (N)<ˆs.
Alternatively iterating ℵ2times rational perfect forcing and Roslanowski
Shelah forcing over a model of CH we obtain ℵ1=non (M)<non (N) =
d=ℵ2. Therefore, it is consistent that
(30) cov (M)<ˆs.
Finally in the iterated Sachs model we have that cof (N) = ℵ1. Hence, it
is consistent with ZFC that
(31) cof (N)<ˆs.
Open problem. Is there any relation between
a) ˆs and cof (M)?
b) ˆs and non (N)?
c) ˆs and cof (N)?
d) ˆs and d?
References
[1] T. Bartoszyński, H Judah, On the structure of the real line, Taylor and Francis,
1995.
[2] A. Blass, T. Hyttinen, Y. Zhang, Mad families and their neighbours,
https://pdfs.semanticscholar.org/6cc7/efe9310c71b9ae107b113ebe3af601d44f32.pdf
[access 7.10.2018].
[3] L. Bukovsky, The structure of the real line, Birkhäuser Basel, 2011.
[4] B. A. Eﬁmov, Diaditcheskie bikompakty, (in Russian), Trudy Mosk. Matem. Ova
14 (1965), 211247.
[5] D. H. Fremlin,Cichoń diagram, Publ. Math. Univ. Pierre Marie Curie 66, Sémin.
Initiation Anal. 23éme Année1983/84, Exp. 5 (1984), 13.
[6] J. Jureczko, On inequalities among some cardinal invariants, Mathematica Aeterna,
Vol. 6, 2016, no. 1, 87  98.
[7] J. Jureczko, Strong sequences and partition relations, Ann. Univ. Paedagog. Crac.
Stud. Math. 16 (2017), 51  59.
[8] J. Jureczko, κstrong sequences and the existence of generalized independent families.
Open Math. 15 (2017), 1277–1282.
34 J. JURECZKO
[9] J. Jureczko, On Banach and Kuratowski theorem, KLusin sets and strong sequences.
Open Math. 16 (2018), 724–729.
Received: October 2018
Joanna Jureczko
Wrocław University of Science and Technology,
Faculty of Electronics
Janiszewskiego Street 11/17, 50372 Wrocław
Email address:joanna.jureczko@pwr.edu.pl
Scientific Issues
Jan Długosz University
in Częstochowa
Mathematics XXIII (2018)
35–44
DOI http://dx.doi.org/10.16926/m.2018.23.03
ON F(p, n)FIBONACCI BICOMPLEX NUMBERS
MIROSŁAW LIANA, ANETTA SZYNALLIANA, AND IWONA WŁOCH
Abstract
In this paper we introduce F(p, n)Fibonacci bicomplex numbers and L(p, n)Lucas
bicomplex numbers as a special type of bicomplex numbers. We give some their properties
and describe relations between them.
1. Introduction
Let consider the set Cof complex numbers a+bi, where a, b ∈R, with
the imaginary unit i. Let Bbe the set of bicomplex numbers wof the form
(1) w=z1+z2j,
where z1, z2∈C.Then iand jare commuting imaginary units, i.e.
(2) ij =ji, i2=j2=−1.
Let w1= (a1+b1i)+(c1+d1i)jand w2= (a2+b2i)+(c2+d2i)j
be arbitrary two bicomplex numbers. Then the equality, the addition, the
substraction, the multiplication and the multiplication by scalar are deﬁned
in the following way.
Equality: w1=w2only if a1=a2, b1=b2, c1=c2, d1=d2,
addition: w1+w2= ((a1+a2)+(b1+b2)i) + ((c1+c2)+(d1+d2)i)j,
substraction: w1−w2= ((a1−a2) + (b1−b2)i) + ((c1−c2) + (d1−d2)i)j,
multiplication by scalar s∈R:sw1= (sa1+sb1i)+(sc1+sd1i)j,
multiplication:
w1·w2= ((a1a2−b1b2−c1c2+d1d2)+(a1b2+a2b1−c1d2−c2d1)i) +
+ ((a1c2+a2c1−b1d2−b2d1)+(a1d2+a2d1+b1c2+b2c1)i)j.
•Mirosław Liana — email: mliana@prz.edu.pl
Rzeszow University of Technology.
•Anetta SzynalLiana — email: aszynal@prz.edu.pl
Rzeszow University of Technology.
•Iwona Włoch — email: iwloch@prz.edu.pl
Rzeszow University of Technology.
36 M. LIANA, A. SZYNALLIANA, AND I. WŁOCH
The bicomplex numbers were introduced in 1892 by Segre, see [5]. The
theory of bicomplex numbers is developed, many of papers concerning this
topic are published quite recently, see for example [2], [3], [4].
The Fibonacci numbers Fnare deﬁned by the recurrence relation Fn=
Fn−1+Fn−2,for n≥2with F0=F1= 1.The nth Lucas number Lnis
deﬁned recursively by Ln=Ln−1+Ln−2for n≥2with the initial terms
L0= 2, L1= 1.
In this paper we recall some generalizations of Fibonacci numbers and
Lucas numbers and we introduce the bicomplex numbers related with these
generalizations.
2. The F(p, n)Fibonacci numbers
The Fibonacci sequence has been generalized in many ways but a very
natural is ﬁrstly to use oneparameter generalization of the Fibonacci se
quence. A generalization uses one parameter p,p≥2was introduced
and studied by Kwaśnik and I. Włoch in the context of the number of
pindependent sets in graphs, see [1]. We recall this deﬁnition.
Let p≥2be integer. Then
(3) F(p, n) = n+ 1,for n= 0,1, . . . , p −1,
F(p, n) = F(p, n −1) + F(p, n −p),for n≥p,
is the F(p, n)Fibonacci number.
Moreover L(p, n)Lucas number is a cyclic version of F(p, n)deﬁned in
the following way
(4) L(p, n) = n+ 1,for n= 0,1,...,2p−1,
L(p, n) = L(p, n −1) + L(p, n −p),for n≥2p,
where p≥2, n ≥0.
Note that for n≥0we have that F(2, n) = Fn+1 and for n≥2L(2, n) =
Ln.
The following Tables present the initial words of the generalized Fibonacci
numbers and the generalized Lucas numbers for special case of nand p.
n0 1 2 3 4 5 6 7 8 9 10
Fn1 1 2 3 5 8 13 21 34 55 89
F(2, n) 1 2 3 5 8 13 21 34 55 89 144
F(3, n) 1 2 3 4 6 9 13 19 28 41 60
F(4, n) 1 2 3 4 5 7 10 14 19 26 36
F(5, n) 1 2 3 4 5 6 8 11 15 20 26
Table 1. The values of F(p, n)and Fn.
ON F(p, n)FIBONACCI BICOMPLEX NUMBERS 37
n0 1 2 3 4 5 6 7 8 9 10
Ln2 1 3 4 7 11 18 29 47 76 123
L(2, n) 1 2 3 4 7 11 18 29 47 76 123
L(3, n) 1 2 3 4 5 6 10 15 21 31 46
L(4, n) 1 2 3 4 5 6 7 8 13 19 26
Table 2. The values of L(p, n)and Ln.
Generalized Fibonacci numbers F(p, n)and generalized Lucas numbers
L(p, n)have been studied recently, mainly with respect to their graph and
combinatorial properties, see for example [7], [8], [9], [10]. Among other
some identities for F(p, n)and L(p, n)were given. We recall some of them.
Theorem 1 ([8]).Let p≥2be integer. Then for n≥p+ 1
(5)
n−p
X
l=0
F(p, l) = F(p, n)−p.
Theorem 2 ([8]).Let p≥2, n ≥pbe integers. Then
(6)
n
X
l=1
F(p, lp −1) + 1 = F(p, np).
Theorem 3 ([6]).Let p≥2, n ≥pbe integers. Then
(7)
n
X
l=1
F(p, lp) = F(p, np + 1) −F(p, 1),
(8)
n
X
l=1
F(p, lp + 1) = F(p, np + 2) −F(p, 2),
(9)
n
X
l=1
F(p, lp + 2) = F(p, np + 3) −F(p, 3).
Theorem 4 ([8]).Let p≥2, n ≥2p−2be integers. Then
(10) F(p, n) =
p−1
X
l=0
F(p, n −(p−1) −l).
Theorem 5 ([8]).Let p≥2, n ≥2pbe integers. Then
(11)
n
X
l=2
L(p, pl) = L(p, np + 1) −(p+ 2).
38 M. LIANA, A. SZYNALLIANA, AND I. WŁOCH
Theorem 6 ([6]).Let p≥2, n ≥2pbe integers. Then
(12)
n
X
l=2
L(p, pl + 1) = L(p, np + 2) −L(p, p + 2).
(13)
n
X
l=2
L(p, pl + 2) = L(p, np + 3) −L(p, p + 3).
(14)
n
X
l=2
L(p, pl + 3) = L(p, np + 4) −L(p, p + 4).
Theorem 7 ([8]).Let p≥2, n ≥2pbe integers. Then
(15) L(p, n) = pF (p, n −(2p−1)) + F(p, n −p).
3. The F(p, n)Fibonacci bicomplex numbers
Let n≥0be an integer. The nth F(p, n)Fibonacci bicomplex number
BF p
nand the nth L(p, n)Lucas bicomplex number BLp
nare deﬁned as
(16) BF p
n= (F(p, n) + F(p, n + 1)i)+(F(p, n + 2) + F(p, n + 3)i)j,
(17) BLp
n= (L(p, n) + L(p, n + 1)i)+(L(p, n + 2) + L(p, n + 3)i)j,
respectively.
Using the above deﬁnitions we can write selected F(p, n)Fibonacci bi
complex numbers, i.e.
BF 3
0= (1 + 2i) + (3 + 4i)j,
BF 3
1= (2 + 3i) + (4 + 6i)j,
BF 3
2= (3 + 4i) + (6 + 9i)j,
. . .
BF 4
0= (1 + 2i) + (3 + 4i)j,
BF 4
1= (2 + 3i) + (4 + 5i)j,
BF 4
2= (3 + 4i) + (5 + 7i)j,
. . .
BF 5
0= (1 + 2i) + (3 + 4i)j,
BF 5
1= (2 + 3i) + (4 + 5i)j,
BF 5
2= (3 + 4i) + (5 + 6i)j,
. . .
In the same way one can easily write selected L(p, n)Lucas bicomplex
numbers.
The addition, the subtraction and the multiplication of F(p, n)Fibonacci
bicomplex numbers and L(p, n)Lucas bicomplex numbers are deﬁned in the
same way as for bicomplex numbers.
ON F(p, n)FIBONACCI BICOMPLEX NUMBERS 39
In the set C,the complex conjugate of x+yi is x+yi =x−yi. In the set
B,for a bicomplex number w= (a+bi) + (c+di)j, there are three distinct
conjugations.
Let BF p
nbe the nth F(p, n)Fibonacci bicomplex number, i.e.
BF p
n= (F(p, n) + F(p, n + 1)i)+(F(p, n + 2) + F(p, n + 3)i)j,
The bicomplex conjugation of BF p
nwith respect to ihas the form
BF p
n
i= (F(p, n) + F(p, n + 1)i) + (F(p, n + 2) + F(p, n + 3)i)j=
= (F(p, n)−F(p, n + 1)i)+(F(p, n + 2) −F(p, n + 3)i)j.
The bicomplex conjugation of BF p
nwith respect to jhas the form
BF p
n
j= (F(p, n) + F(p, n + 1)i)−(F(p, n + 2) + F(p, n + 3)i)j=
= (F(p, n) + F(p, n + 1)i)+(−F(p, n + 2) −F(p, n + 3)i)j.
The third kind of conjugation is a composition of the above two conjuga
tions. Putting k:= ji =ij we can deﬁne the bicomplex conjugation of BF p
n
with respect to kas follows
BF p
n
k= (F(p, n) + F(p, n + 1)i)−(F(p, n + 2) + F(p, n + 3)i)j=
= (F(p, n)−F(p, n + 1)i)+(−F(p, n + 2) + F(p, n + 3)i)j.
Using the bicomplex conjugation of BF p
nwith respect to i, j, k respec
tively and (16) we can write
BF p
n·BF p
n
i=
=F(p, n) + F(p, n + 1)i2− F(p, n + 2) + F(p, n + 3)i2+
+2<(F(p, n) + F(p, n + 1)i)·(F(p, n + 2) + F(p, n + 3)i)j=
= (F(p, n))2+ (F(p, n + 1))2−(F(p, n + 2))2−(F(p, n + 3))2+
+2 (F(p, n)F(p, n + 2) + F(p, n + 1)F(p, n + 3)) j.
BF p
n·BF p
n
j=
= (F(p, n) + F(p, n + 1)i)2+ (F(p, n + 2) + F(p, n + 3)i)2=
= (F(p, n))2−(F(p, n + 1))2+ (F(p, n + 2))2−(F(p, n + 3))2+
+2 (F(p, n)F(p, n + 1) + F(p, n + 2)F(p, n + 3)) i.
BF p
n·BF p
n
k=
=F(p, n) + F(p, n + 1)i2+F(p, n + 2) + F(p, n + 3)i2+
−2=(F(p, n) + F(p, n + 1)i)·(F(p, n + 2) + F(p, n + 3)i)k=
= (F(p, n))2+ (F(p, n + 1))2+ (F(p, n + 2))2+ (F(p, n + 3))2+
−2 (F(p, n + 1)F(p, n + 2) −F(p, n)F(p, n + 3)) k.
In the set C,the modulus of x+yi is x+yi=q(x+yi)·(x+yi) =
px2+y2.In the set Bthere are four diﬀerent moduli, named: real modulus
40 M. LIANA, A. SZYNALLIANA, AND I. WŁOCH
BF p
n,i−modulus B F p
ni,j−modulus B F p
njand k−modulus B F p
nk.We
give the formulae of the squares of these modules:
BF p
n2=F(p, n) + F(p, n + 1)i2+F(p, n + 2) + F(p, n + 3)i2=
= (F(p, n))2+ (F(p, n + 1))2+ (F(p, n + 2))2+ (F(p, n + 3)2,
BF p
n2
i=BF p
n·BF p
n
i,
BF p
n2
j=BF p
n·BF p
n
j,
BF p
n2
k=BF p
n·BF p
n
k.
The diﬀerent conjugations and squares of modules for L(p, n)Lucas bi
complex number BLp
nare presented as follows
BLp
n
i= (L(p, n)−L(p, n + 1)i)+(L(p, n + 2) −L(p, n + 3)i)j,
BLp
n
j= (L(p, n) + L(p, n + 1)i)+(−L(p, n + 2) −L(p, n + 3)i)j,
BLp
n
k= (L(p, n)−L(p, n + 1)i)+(−L(p, n + 2) + L(p, n + 3)i)j.
BLp
n2= (L(p, n))2+ (L(p, n + 1))2+ (L(p, n + 2))2+ (L(p, n + 3)2,
BLp
n2
i= (L(p, n))2+ (L(p, n + 1))2−(L(p, n + 2))2−(L(p, n + 3))2+
+2 (L(p, n)L(p, n + 2) + L(p, n + 1)L(p, n + 3)) j.
BLp
n2
j= (L(p, n))2−(L(p, n + 1))2+ (L(p, n + 2))2−(L(p, n + 3))2+
+2 (L(p, n)L(p, n + 1) + L(p, n + 2)L(p, n + 3)) i.
BLp
n2
k= (L(p, n))2+ (L(p, n + 1))2+ (L(p, n + 2))2+ (L(p, n + 3))2+
−2 (L(p, n + 1)L(p, n + 2) −L(p, n)L(p, n + 3)) k.
4. Properties of F(p, n)Fibonacci bicomplex numbers
We will give some properties of F(p, n)Fibonacci bicomplex numbers
and L(p, n)Lucas bicomplex numbers.
Theorem 8. Let p≥2be integer. Then for n≥p+ 1
(18)
n−p
P
l=0
BF p
l=BF p
n−[p+ (p+F(p, 0))i+
+ ((p+F(p, 0) + F(p, 1)) + (p+F(p, 0) + F(p, 1) + F(p, 2))i)j].
ON F(p, n)FIBONACCI BICOMPLEX NUMBERS 41
Proof. Using (5) and (16) we have
n−p
P
l=0
BF p
l=BF p
0+BF p
1+. . . +B F p
n−p=
= (F(p, 0) + F(p, 1)i)+(F(p, 2) + F(p, 3)i)j+
+(F(p, 1) + F(p, 2)i)+(F(p, 3) + F(p, 4)i)j+. . . +
+(F(p, n −p) + F(p, n −p+ 1)i)+
+(F(p, n −p+ 2) + F(p, n −p+ 3)i)j=
=F(p, 0) + F(p, 1) + . . . +F(p, n −p)+
+ (F(p, 1) + . . . +F(p, n −p+ 1) + F(p, 0) −F(p, 0)) i+
+ [F(p, 2) + . . . +F(p, n −p+ 2) + F(p, 0) + F(p, 1) −F(p, 0)+
−F(p, 1) + (F(p, 3) + . . . +F(p, n −p+ 3) + F(p, 0) + F(p, 1)+
+F(p, 2) −F(p, 0) −F(p, 1) −F(p, 2)) i]j=
= (F(p, n)−p+ (F(p, n + 1) −p−F(p, 0))i) +
+ [(F(p, n + 2) −p−F(p, 0) −F(p, 1)) +
+ (F(p, n + 3) −p−F(p, 0) −F(p, 1) −F(p, 2)) i]j=
=BF p
n−(p+ (p+F(p, 0)) i)−[(p+F(p, 0) + F(p, 1))+
+ (p+F(p, 0) + F(p, 1) + F(p, 2)) i]j,
which ends the proof.
Theorem 9. Let p≥2, n ≥pbe integers. Then
(19)
n
X
l=1
BF p
lp−1=BF p
np −[(F(p, 0) + F(p, 1)i)+(F(p, 2) + F(p, 3)i)j].
Proof. Using (16) we have
n
P
l=1
BF p
lp−1=BF p
p−1+BF p
2p−1+. . . +B F p
np−1=
= (F(p, p −1) + F(p, p)i)+(F(p, p + 1) + F(p, p + 2)i)j+
+(F(p, 2p−1) + F(p, 2p)i)+(F(p, 2p+ 1) + F(p, 2p+ 2)i)j+. . . +
+(F(p, np −1) + F(p, np)i)+(F(p, np + 1) + F(p, np + 2)i)j=
=F(p, p −1) + F(p, 2p−1) + . . . +F(p, np −1)+
+ (F(p, p) + F(p, 2p) + . . . +F(p, np)) i+
+ [(F(p, p + 1) + F(p, 2p+ 1) + . . . +F(p, np + 1)) +
+ (F(p, p + 2) + F(p, 2p+ 2) + . . . +F(p, np + 2)) i]j.
Writing (6) as
n
P
l=1
F(p, lp −1) = F(p, np)−1 = F(p, np)−F(p, 0) and using
(7)–(9) we obtain (19).
Theorem 10. Let p≥2, n ≥2p−2be integers. Then
(20) BF p
n=
p−1
X
l=0
BF p
n−(p−1)−l.
42 M. LIANA, A. SZYNALLIANA, AND I. WŁOCH
Proof. Using (10) and (16) we have
p−1
P
l=0
BF p
n−(p−1)−l=BF p
n−(p−1) +BF p
n−(p−1)−1+. . . +B F p
n−(p−1)−(p−1) =
= (F(p, n −(p−1)) + F(p, n −(p−1) + 1)i)+
+ [F(p, n −(p−1) + 2) + F(p, n −(p−1) + 3)i]j+
+(F(p, n −(p−1) −1) + F(p, n −(p−1))i)+
+ [F(p, n −(p−1) + 1) + F(p, n −(p−1) + 2)i]j+. . . +
+(F(p, n −(p−1) −(p−1)) + F(p, n −(p−1) −(p−1) + 1)i)+
+ [F(p, n −(p−1) −(p−1) + 2) + F(p, n −(p−1) −(p−1) + 3)i]j=
= (F(p, n) + F(p, n + 1)i)+(F(p, n + 2) + F(p, n + 3)i)j=B F p
n,
which ends the proof.
Theorem 11. Let p≥2, n ≥2pbe integers. Then
(21)
n
X
l=2
BLp
pl =BLp
np+1 −BLp
p+1.
Proof. Using (17) we have
n
P
l=2
BLp
pl =BLp
2p+BLp
3l+. . . +BLp
nl =
= (L(p, 2p) + L(p, 2p+ 1)i)+(L(p, 2p+ 2) + L(p, 2p+ 3)i)j+
+(L(p, 3p) + L(p, 3p+ 1)i)+(L(p, 3p+ 2) + L(p, 3p+ 3)i)j+. . . +
+(L(p, np) + L(p, np + 1)i)+(L(p, np + 2) + L(p, np + 3)i)j+
=L(p, 2p) + L(p, 3p) + . . . +L(p, np)+
+ (L(p, 2p+ 1) + L(p, 3p+ 1) + . . . +L(p, np + 1)) i+
+ [(L(p, 2p+ 2) + L(p, 3p+ 2) + . . . +L(p, np + 2)) +
+ (L(p, 2p+ 3) + L(p, 3p+ 3) + . . . +L(p, np + 3)) i]j.
Writing (11) as
n
P
l=2
L(p, pl) = L(p, np + 1) −L(p, p + 1) and using (12)–(14)
we obtain (21).
Theorem 12. Let p≥2, n ≥2pbe integers. Then
(22) BLp
n=p·BF p
n−(2p−1) +BF p
n−p.
Proof. Using (16) we have
BF p
n−(2p−1) = (F(p, n −(2p−1)) + F(p, n −(2p−1) + 1)i)+
+(F(p, n −(2p−1) + 2) + F(p, n −(2p−1) + 3)i)j
and
BF p
n−p= (F(p, n −p) + F(p, n −p+ 1)i)+
+(F(p, n −p+ 2) + F(p, n −p+ 3)i)j,
ON F(p, n)FIBONACCI BICOMPLEX NUMBERS 43
consequently
p·BF p
n−(2p−1) +BF p
n−p=
=p·F(p, n −(2p−1)) + F(p, n −p)+
+ (p·F(p, (n+ 1) −(2p−1)) + F(p, (n+ 1) −p)) i+
+ [(p·F(p, (n+ 2) −(2p−1)) + F(p, (n+ 2) −p)) +
+ (p·F(p, (n+ 3) −(2p−1)) + F(p, (n+ 3) −p)) i]j
Using (15) we have
p·BF p
n−(2p−1) +BF p
n−p=
= (L(p, n) + L(p, n + 1)i)+(L(p, n + 2) + L(p, n + 3)i)j,
which ends the proof.
For integers p, n, l, p ≥2, n ≥2,0≤l≤nwe have (see [9]) the direct
formula for F(p, n)Fibonacci number
F(p, n) = X
l≥0
f(p, n, l),
where
f(p, n, l) = n−(p−1)(l−1)
l.
Using this direct formula other forms of given earlier identities can be
given.
References
[1] M. Kwaśnik, I. Włoch, The total number of generalized stable sets and kernels of
graphs, Ars Combinatoria 55 (2000), 139146.
[2] M.E. LunaElizarrarás, M. Shapiro, D.C. Struppa, A. Vajiac, Bicomplex Holo
morphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers,
Birkhäuser (2015).
[3] M.E. LunaElizarrarás, M. Shapiro, D.C. Struppa, A. Vajiac, Bicomplex Numbers
and their Elementary Functions, CUBO 14(2) (2012), 6180.
doi: http://dx.doi.org/10.4067/S071906462012000200004
[4] D. Rochon, M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers,
Analele Universităţii Oradea, Fascicola Matematica 11 (2004), 71110.
[5] C. Segre, Le Rappresentazioni Reali delle Forme Complesse a Gli Enti Iperalgebrici,
Mathematische Annalen 40 (1892), 413467.
doi: https://doi.org/10.1007/BF01443559
[6] A. SzynalLiana, I. Włoch, On F(p, n)Fibonacci Quaternions, submitted.
[7] A. Włoch, On generalized Fibonacci numbers and kdistance Kpmatchings in graphs,
Discrete Applied Mathematics 160 (2012), 13991405.
doi: https://doi.org/10.1016/j.dam.2012.01.008
[8] A. Włoch, Some identities for the generalized Fibonacci numbers and the generalized
Lucas numbers, Applied Mathematics and Computation 219 (2013), 55645568.
doi: https://doi.org/10.1016/j.amc.2012.11.030
44 M. LIANA, A. SZYNALLIANA, AND I. WŁOCH
[9] A. Włoch, Some interpretations of the generalized Fibonacci numbers, AKCE Inter
national Journal of Graphs and Combinatorics 9(2) (2012), 123133.
[10] I. Włoch, Generalized Fibonacci polynomial of graph, Ars Combinatoria 68 (2003),
4955.
Received: June 2018
Mirosław Liana
Rzeszow University of Technology,
The Faculty of Management,
al. Powstańców Warszawy 10, 35959 Rzeszów, Poland
Ema