On the upfamily extension of a doppelsemigroup

  • V. M. Gavrylkiv Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Keywords: semigroup, upfamily, doppelsemigroups

Abstract

A family $\mathcal{U}$ of non-empty subsets of a set $D$ is called  an upfamily if for each set $U\in\mathcal{U}$ any set $F\supset U$ belongs to $\mathcal{U}$. 
The upfamily extension $\upsilon(D)$ of $D$ consists of all upfamilies on $D$.
Any associative binary operation $* \colon D\times D \to D$ can be extended to an associative binary operation $$*:\upsilon(D)\times \upsilon(D)\to \upsilon(D), \ \ \ \mathcal U*\mathcal V=\big\langle\bigcup_{a\in
U}a*V_a:U\in\mathcal U,\;\;\{V_a\}_{a\in U}\subset\mathcal V\big\rangle.$$
In the paper, we show that the upfamily extension $(\upsilon(D),\dashv,\vdash)$ of a (strong) doppelsemigroup  $(D,\dashv,\vdash)$ is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category $\mathbf {DSG}$ whose objects are doppelsemigroups and  morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup $(D,\dashv, \vdash)$ of cardinality $|D|\geq 2$ contains a subgroup, isomorphic to $C_2\times \mathrm{Aut\mkern 2mu}(D,\dashv, \vdash)$. Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups.

Author Biography

V. M. Gavrylkiv, Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

Vasyl Stefanyk Precarpathian National University

Ivano-Frankivsk, Ukraine



References

T. Banakh, V. Gavrylkiv, Algebra in superextension of groups, II: cancelativity and centers, Algebra Discrete Math., 4 (2008), 1–14.

T. Banakh, V. Gavrylkiv, Algebra in superextension of groups: minimal left ideals, Mat. Stud., 31 (2009), №2, 142–148.

T. Banakh, V. Gavrylkiv, Extending binary operations to functor-spaces, Carpathian Math. Publ., 1 (2009), №2, 113–126.

T. Banakh, V. Gavrylkiv, Algebra in superextensions of semilattices, Algebra Discrete Math., 13 (2012), №1, 26–42.

T. Banakh, V. Gavrylkiv, Characterizing semigroups with commutative superextensions, Algebra Discrete Math., 17 (2014), №2, 161–192.

T. Banakh, V. Gavrylkiv, On structure of the semigroups of k-linked upfamilies on groups, Asian-European J. Math., 10 (2017), №4, 1750083[15 pages] doi: 10.1142/S1793557117500838.

T. Banakh, V. Gavrylkiv, Automorphism groups of superextensions of groups, Mat. Stud., 48 (2017), №2, 134–142. doi: 10.15330/ms.48.2.134-142

T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutativity, Algebra Discrete Math., 3 (2008), 1–29.

S.J. Boyd, M. Gould, A. Nelson, Interassociativity of Semigroups, In: Proceedings of the Tennessee Topology Conference, 1997, World Scientific, 33–51.

R. Dedekind, Uber Zerlegungen von Zahlen durch ihre grussten gemeinsammen Teiler, In: Gesammelte Werke, 1897, V.1, Springer, 103–148.

M. Drouzy, La structuration des ensembles de semigroupes d’ordre 2, 3 et 4 par la relation d’interassociativit´e, 1986, manuscript.

V. Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud., 28 (2007), №1, 92–110.

V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces, Mat. Stud., 29 (2008), №1, 18–34.

V. Gavrylkiv, Semigroups of centered upfamilies on groups, Lobachevskii J. Math., 38 (2017), №3, 420–428. doi: 10.1134/S1995080217030106.

V. Gavrylkiv, Superextensions of three-element semigroups, Carpathian Math. Publ., 9 (2017), №1, 28–36. doi: 10.15330/cmp.9.1.28-36

V. Gavrylkiv, On the automorphism group of the superextension of a semigroup, Mat. Stud., 48 (2017), №1, 3–13. doi: 10.15330/ms.48.1.3-13

V. Gavrylkiv, Automorphisms of semigroups of k-linked upfamilies, J. Math. Sci., , 234 (2018), №1, 21–34. doi: 10.1007/s10958-018-3978-7

V. Gavrylkiv, Automorphism groups of semigroups of upfamilies, Asian-European J. Math., 13 (2020), №1, 2050099. doi: 10.1142/S1793557120500990

V.M. Gavrylkiv, D.V. Rendziak, Interassociativity and three-element doppelsemigroups, Algebra Discrete Math., 28 (2019), №2, 224–247.

V.M. Gavrylkiv, Note on cyclic doppelsemigroups, Algebra Discrete Math., 34 (2022), №1, 15–21. doi:10.12958/adm1991

M. Gould, K.A. Linton, A.W. Nelson, Interassociates of monogenic semigroups, Semigroup Forum, 68 (2004), 186–201. doi: 10.1007/s00233-002-0028-y

M. Gould, R.E. Richardson, Translational hulls of polynomially related semigroups, Czechoslovak Math. J., 33 (1983), 95–100.

J.B. Hickey, Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc., 26 (1983), 371–382.

J.B. Hickey, On Variants of a semigroup, Bull. Austral. Math. Soc., 34 (1986), 447–459.

N. Hindman, D. Strauss, Algebra in the Stone-ˇCech compactification, 1998, de Gruyter: Berlin, New York.

J.M. Howie, Fundamentals of semigroup theory, 1995, Oxford University Press, New York.

J.L. Loday, Dialgebras. In: Dialgebras and related operads: Lect. Notes Math., 2001, V.1763, Berlin: Springer-Verlag, 7–66.

T. Pirashvili, Sets with two associative operations, Cent. Eur. J. Math., 2 (2003), 169–183. doi: 10.2478/BF02476006

B. Richter, Dialgebren, Doppelalgebren und ihre Homologie., 1997, Diplomarbeit, Universitat Bonn.

B.M. Schein, Restrictive semigroups and bisemigroups., 1989, Technical Report. University of Arkansas, Fayetteville, Arkansas, USA, 1–23.

B.M. Schein, Restrictive bisemigroups, Izv. Vyssh. Uchebn. Zaved. Mat., 1 (1965), №44, 168–179. (in Russian)

A. Teleiko, M. Zarichnyi, Categorical topology of compact Hausdofff spaces, 1999, Lviv: VNTL.

A.V. Zhuchok, M. Demko, Free n-dinilpotent doppelsemigroups, Algebra Discrete Math., 22 (2016), №2, 304–316.

A.V. Zhuchok, Free products of doppelsemigroups, Algebra Univers., 77 (2017) №3, 361–374. doi: 10.1007/s00012-017-0431-6

A.V. Zhuchok, Free left n-dinilpotent doppelsemigroups, Commun. Algebra, 45 (2017), №11, 4960–4970. doi: 0.1080/00927872.2017.1287274

A.V. Zhuchok, Structure of free strong doppelsemigroups, Commun. Algebra, 46 (2018), №8, 3262–3279. doi: 10.1080/00927872.2017.1407422

A.V. Zhuchok, Relatively free doppelsemigroups. Monograph series Lectures in Pure and Applied Mathematics, 2018, V.5, Germany, Potsdam: Potsdam University Press, 86 p.

Y.V. Zhuchok, J. Koppitz, Representations of ordered doppelsemigroups by binary relations, Algebra Discrete Math., 27 (2019), №1, 144–154.

A.V. Zhuchok, Yul. V. Zhuchok, J. Koppitz, Free rectangular doppelsemigroups, J. Algebra Appl., 19 (2020), №11, 2050205. doi: 10.1142/S0219498820502059.

D. Zupnik, On interassociativity and related questions, Aequationes Math., 6 (1971), 141–148.

Published
2024-06-19
How to Cite
Gavrylkiv, V. M. (2024). On the upfamily extension of a doppelsemigroup. Matematychni Studii, 61(2), 123-135. https://doi.org/10.30970/ms.61.2.123-135
Section
Articles