We find faithful representations of a finite unar (an algebra with one unary operation on a finite set) in some standard constructions. We prove that every finite unar can be faithfully represented by the residues modulo $n$ with the operation $f(x)= x\cdot a \,\mod n$ for suitable $n$ and $a$. Besides, for every integer $d\ge 2$, there exists a faithful representation of every finite unar by residues modulo $n$ with the operation $f(x)= x^d \,\mod n$ for suitable $n$. Further, for any $d\ge 3$, every finite unar can be faithfully presented by invertible residues modulo $n$ with the operation $f(x)= x^d \,\mod n$ for suitable $n$. (The later assertion is not true for $d=2$).
Keywords: representations of unar
Received September 25, 2024
Revised February 11, 2025
Accepted February 17, 2025
Funding Agency: This work was supported by the Russian Science Foundation (project no. 22-11-00052).
Igor Borisovich Kozhukhov, Dr. Phys.-Math. Sci., Prof., Nat. Res. Univ. MIET; Fac. of Mech. and Math. of Moscow State Univ.; Russian Presidental Academy of Nat. Econom. and Public Admin., Moscow, Russia, e-mail: kozhuhov_i_b@mail.ru
Vladimir Alexandrovich Letsko, Cand. Sci. (Pedagog.), Volgograd State Socio-pedagogical University Volgograd, Russia, e-mail: val-etc@yandex.ru
REFERENCES
1. Kargapolov M.I., Merzljakov J.I. Fundamentals of the theory of groups. Ser. Graduate texts in mathematics. New York, Springer, 1979, 221 p., ISBN-10: 1461299667 . Original Russian text published in Kargapolov M.I., Merzljakov J.I. Osnovy teorii grupp, Moscow, Nauka Publ., 1977, 240 p.
2. Clifford A.H., Preston G.B. The algebraic theory of semigroups. Math. Surv., no. 7, Providence, Rhode Island, Amer. Math. Soc., vol. I, 1961, 244 p. ISBN: 9780821802717 ; vol. II, 1967, 352 p., ISBN: 9780821802724 . Translated to Russian under the title Algebraicheskaya teoriya polugrupp, Moscow, Mir Publ., 1972, vol. 1, 286 p.; vol. 2, 423 p.
3. Curtis C.W., Reiner I. Representation theory of finite groups and associative algebras. AMS Chelsea Publ. Ser., vol. 356, Amer. Math. Soc., 1966, 689 p. ISBN: 0821869450 . Translated to Russian under the title Teoriya predstavleniy konechnykh grupp i assotsiativnykh algebr, Moscow, Nauka Publ., 1969, 668 p.
4. Steinberg B. Representation theory of finite monoids. Cham, Springer, 2016, 317 p. https://doi.org/10.1007/978-3-319-43932-7
5. Whitman P.M. Lattices, equivalence relations, and subgroups. Bull. Amer. Math. Soc., 1946, vol. 52, no. 6, pp. 507–522.
6. Kozhukhov I.B., Mikhalev A.V. Acts over semigroups. J. Math. Sci., 2023, vol. 269, no. 3, pp. 362–401. https://doi.org/10.1007/s10958-023-06287-3
7. Zelinka B. Graphs of semigroups. Časopis pro pěstování mat., 1981, vol. 106, no. 4, pp. 407–408. https://doi.org/10.21136/CPM.1981.108493
8. Lucheta C., Miller E., Reiter C. Digraphs from powers modulo p. Fibonacci Q., 1996, vol. 34, no. 3, pp. 226–239. https://doi.org/10385/f4752j454
9. Wilson B. Power digraphs modulo n. Fibonacci Q., 1998, vol. 36, no. 3, pp. 229–239.
10. Min Sha. On the cycle structure of repeated expontiation modulo a prime power. Fibonacci Q., 2011, vol. 49, no. 4, pp. 340–347. https://doi.org/10.1080/00150517.2011.12428034
11. Somer L., Křížek M. The structure of digraphs associated with the congruence $x^k \equiv y \pmod n)$. Czech. Math. J., 2011, vol. 61, no. 2, pp. 337–358. https://doi.org/ 10.1007/s10587-011-0079-x
12. Martin G., Pomerance C.B. The iterated Carmichael λ-function and the number of cycles of the power generator. Acta Arithmetica, 2005, vol. 118, no. 4, pp. 305–335. https://doi.org/ 10.4064/aa118-4-1
13. Kurlberg P., Pomerance C.B. On the periods of the linear congruential and power generators. Acta Arithmetica, 2005, vol. 119, no. 2, pp. 149–169. https://doi.org/10.4064/aa119-2-2
14. Parker E.T. On multiplicative semigroups of residue classes. Proc. Amer. Math. Soc., 1954, vol. 5, no. 4, pp. 612–616.
15. Slobodskoy G., Letsko V.A. On the representation of finite unars in $\mathbb Z_n$. In: Vestnik SNO: sbornik statei / Volgograd. gos. pedagog. un-t. Ser, “Matematika i Tekhnika”. No 7. Volgograd: Izd-vo “Peremena”, 1995. P. 3–6 (in Russian).
16. Borevich Z.I., Shafarevich I.R. Number theory. Orlando, Florida, Academic Press Inc., 1986, 435 p. ISBN-13: 978-0121178512 . Original Russian text was published in Borevich Z.I., Shafarevich I.R. Teoriya chisel, Moscow, Nauka Publ., 1985, 504 p.
17. Letsko V.A. Ot zadachi k issledovaniyu [From problem to research], St. Petersburg, SMIO Press, 2021, 336 p. ISBN: 978-5-7704-0368-8 .
Cite this article as: I.B. Kozhukhov, V.A. Letsko. Representation of unars by sets of residues. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 77–89.
