$f$-REPRESENTATIVES GROUPS

Abstract

The $b$-parts, $b$-addition of real numbers and some of their extensions (e.g., $f$-multiplications) were introduced and studied by the author. In this paper, we introduce $f$-representatives groups of a given group $(G,\cdot)$ which can be considered as a generalization of the group of the least non-negative residues (modulo $n$). Thereafter, we study some of their important properties and their relations among the ground group $(G,\cdot)$, a related quotient group, and $f$-grouplikes. Finally, many equivalent conditions for $f$-representatives groups and several examples in the group of real numbers are given.

Keywords

• $b$-Addition
• $f$-multiplication
• $f$-grouplike
• decomposer function

References

1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989.
2. G. M. Bergman, A note on factorizations of finite groups, J. Iran. Math. Soc., 1(2) (2020), 157-161.
3. A. H. Clifford and D. D. Miller, Semigroups having zeroid elements, Amer. J. Math., 70(1) (1948), 117-125.
4. K. Corradi and S. Szabo, Factoring certain infinite abelian groups by cyclic subsets, Pure Math. Appl. Ser. A, 2 (1992), 285-290.
5. G. Hajos, Uber einfache und mehrfache Bedeckung des $n$-dimensionalen Raumes mit einem Wurfelgitter, Math. Z., 47 (1941), 427-467.
6. M. H. Hooshmand, Grouplikes, Bull. Iranian Math. Soc., 39(1) (2013), 65-85.
7. M. H. Hooshmand, New results for the multiplicative symmetric equation and related conjecture, Aequationes Math., 94(1) (2020), 123-136.
8. R. P. Hunter, On the structure of homogroups with applications to the theory of compact connected semigroups, Fund. Math., 52 (1963), 69-102.

9. A. D. Sands, The factorization of abelian groups II, Quart. J. Math. Oxford Ser. (2), 13 (1962), 45-54.
10. S. Schwarz, Semigroups with a universally minimal left ideal, Acta Sci. Math. (Szeged), 52(1-2) (1988), 21-28.
11. S. Szabo and A. D. Sands, Factoring Groups Into Subsets, Lecture Notes in Pure and Applied Mathematics, 257, CRC Press, Boca Raton, FL, 2009.