Algebraic structures in the sets of surjective functions

Bartoszewicz, Artur; Bienias, Marek; Głąb, Szymon; Natkaniec, Tomasz

doi:10.1016/j.jmaa.2016.04.013

Cited by 15 publications

(13 citation statements)

References 30 publications

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“…Although it is not explicitly shown in [8], it can be deduced that ES n ES Ä is 2 -lineable for every < Ä since, in that case, ES n ES Ä ES n ES Ä and, by Theorem 3.10 is 2 -lineable. The case D @ 0 and Ä D c is explicitly studied in [11, Theorem 2.14].…”

Section: Algebraic Size Of Sets Of Surjective Functionsmentioning

confidence: 94%

“…A thorough description of the most relevant lineability problems and other related topics can be found in the monograph [4] or in the expository paper [5]. The interested reader may also consult the references [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

“…However, there are a few nice results in the literature about algebras of surjective functions in C C . We refer to [8] for a complete account of results on algebrability of complex surjective functions. For instance, in [8] it is proved that the family of complex Jones functions is strongly 2 c -algebrable.…”

mentioning

confidence: 99%

“…We refer to [8] for a complete account of results on algebrability of complex surjective functions. For instance, in [8] it is proved that the family of complex Jones functions is strongly 2 c -algebrable. Here a complex Jones function stands for a mapping f 2 C C such that for every closed set C C 2 with uncountable projection on the first coordinate,…”

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confidence: 99%

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A hierarchy in the family of real surjective functions

et al. 2017

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This expository paper focuses on the study of extreme surjective functions in R R . We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, Ä-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in R R .

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Section: Algebraic Size Of Sets Of Surjective Functionsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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A hierarchy in the family of real surjective functions

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“…Now, we will study the algebrability of the family VFC\FC. We apply the method using ultrafilters on N, initiated in [3] and developed in [9]. Define the standard projections π i : R 2 → R (i = 1, 2) by π 1 (x, y) := x, π 2 (x, y) := y.…”

Section: Large Algebraic Structures Within Families Of Feebly Continumentioning

confidence: 99%

Families of feebly continuous functions and their properties

Balcerzak

Natkaniec

Terepeta

2020

Topology and its Applications

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Additivity Coefficients for all Classes in the Algebra of Darboux-Like Maps on $$\mathbb {R}$$

et al. 2020

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The class $$\mathbb D$$ D of generalized continuous functions on $$\mathbb {R}$$ R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property. The algebra $$\mathcal {A}(\mathbb D)$$ A ( D ) of classes of functions generated by these families contains 17 atoms. In this work we will calculate the values of the additivity coefficient $${{\,\mathrm{A}\,}}(\mathcal {F})$$ A ( F ) for all atoms $$\mathcal {F}$$ F in the algebra $$\mathcal {A}(\mathbb D)$$ A ( D ) . We also determine the values $${{\,\mathrm{A}\,}}(\mathcal {F})$$ A ( F ) for a lot of other families $$\mathcal {F}\in \mathcal {A}(\mathbb D)$$ F ∈ A ( D ) . Open questions and new directions of research shall also be provided.

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Algebraic structures in the sets of surjective functions

Cited by 15 publications

References 30 publications

A hierarchy in the family of real surjective functions

A hierarchy in the family of real surjective functions

Families of feebly continuous functions and their properties

Additivity Coefficients for all Classes in the Algebra of Darboux-Like Maps on $$\mathbb {R}$$

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