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**On the lattice generated by Hamel functions.**
*(English)*
Zbl 1259.26002

Recently, K. Płotka in his paper [Proc. Am. Math. Soc. 131, No. 4, 1031–1041 (2003; Zbl 1012.15001)] introduced the notion of a Hamel function. This notion based on the notion of a Hamel basis on \(\mathbb R^2\), where \(\mathbb R\) is the set of reals.

A function \(f : \mathbb R \to \mathbb R\) is a Hamel function (written \(f\in \mathrm{HF}\)) if \(f\) considered as a subset of \(\mathbb R^2\) is a Hamel basis of \(\mathbb R^2\) and it is linearly independent over \(\mathbb Q\) (written \(f\in\mathrm{LIF}\)) if it is linearly independent over \(\mathbb Q\) as a subset of \(\mathbb R^2\), where \(\mathbb Q\) is the set of rationals. If the domain of such a function \(f\) is a subset of \(\mathbb R\) it is named partial function and is denoted by \(f\in\mathrm{PHF}\) or \(f\in\mathrm{PLIF}\) respectively.

The symbols \(\mathcal L(\mathrm{HF})\) and \(\mathcal L(\mathrm{LIF})\) stand for the lattices generated by HF and LIF respectively.

This paper contributes to the further study of the class of Hamel functions and the class of linearly independent functions.

More precisely the author introduces the notion of \(n\)-Hamel functions and proves between other interesting results that \(\mathcal L(\mathrm{HF})=\{f\in\mathbb R^\mathbb R: f\text{ is an }n\)-Hamel function for some

A function \(f : \mathbb R \to \mathbb R\) is a Hamel function (written \(f\in \mathrm{HF}\)) if \(f\) considered as a subset of \(\mathbb R^2\) is a Hamel basis of \(\mathbb R^2\) and it is linearly independent over \(\mathbb Q\) (written \(f\in\mathrm{LIF}\)) if it is linearly independent over \(\mathbb Q\) as a subset of \(\mathbb R^2\), where \(\mathbb Q\) is the set of rationals. If the domain of such a function \(f\) is a subset of \(\mathbb R\) it is named partial function and is denoted by \(f\in\mathrm{PHF}\) or \(f\in\mathrm{PLIF}\) respectively.

The symbols \(\mathcal L(\mathrm{HF})\) and \(\mathcal L(\mathrm{LIF})\) stand for the lattices generated by HF and LIF respectively.

This paper contributes to the further study of the class of Hamel functions and the class of linearly independent functions.

More precisely the author introduces the notion of \(n\)-Hamel functions and proves between other interesting results that \(\mathcal L(\mathrm{HF})=\{f\in\mathbb R^\mathbb R: f\text{ is an }n\)-Hamel function for some

Reviewer: Nikolaos Papanastassiou (Athens)

### MSC:

26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |

15A03 | Vector spaces, linear dependence, rank, lineability |

54C40 | Algebraic properties of function spaces in general topology |

26A21 | Classification of real functions; Baire classification of sets and functions |