Geodesic
Abelian doppelsemigroups
Algebra and discrete mathematics, Tome 26 (2018) no. 2, pp. 290-304.

Voir la notice de l'article provenant de la source Math-Net.Ru

A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain identities. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as doppelalgebras, duplexes, interassociative semigroups, restrictive bisemigroups, dimonoids and trioids. This paper is devoted to the study of abelian doppelsemigroups. We show that every abelian doppelsemigroup can be constructed from a left and right commutative semigroup and describe the free abelian doppelsemigroup. We also characterize the least abelian congruence on the free doppelsemigroup, give examples of abelian doppelsemigroups and find conditions under which the operations of an abelian doppelsemigroup coincide.
Keywords: doppelsemigroup, abelian doppelsemigroup, free abelian doppelsemigroup, free doppelsemigroup, interassociativity, semigroup, congruence
Mots-clés : doppelalgebra. 
@article{ADM_2018_26_2_a7,
     author = {Anatolii V. Zhuchok and Kolja Knauer},
     title = {Abelian doppelsemigroups},
     journal = {Algebra and discrete mathematics},
     pages = {290--304},
     publisher = {mathdoc},
     volume = {26},
     number = {2},
     year = {2018},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/ADM_2018_26_2_a7/}
}
TY  - JOUR
AU  - Anatolii V. Zhuchok
AU  - Kolja Knauer
TI  - Abelian doppelsemigroups
JO  - Algebra and discrete mathematics
PY  - 2018
SP  - 290
EP  - 304
VL  - 26
IS  - 2
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/ADM_2018_26_2_a7/
LA  - en
ID  - ADM_2018_26_2_a7
ER  - 
%0 Journal Article
%A Anatolii V. Zhuchok
%A Kolja Knauer
%T Abelian doppelsemigroups
%J Algebra and discrete mathematics
%D 2018
%P 290-304
%V 26
%N 2
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/ADM_2018_26_2_a7/
%G en
%F ADM_2018_26_2_a7
Anatolii V. Zhuchok; Kolja Knauer. Abelian doppelsemigroups. Algebra and discrete mathematics, Tome 26 (2018) no. 2, pp. 290-304. https://geodesic-test.mathdoc.fr/item/ADM_2018_26_2_a7/

[1] S. J. Boyd, M. Gould, “Interassociativity and isomorphism”, Pure Math. Appl., 10:1 (1999), 23–30 | MR | Zbl

[2] S. J. Boyd, M. Gould, A. W. Nelson, “Interassociativity of semigroups”, Proceedings of the Tennessee Topology Conference (Nashville, TN, USA 1996), World Scientific, Singapore, 1997, 33–51 | MR | Zbl

[3] M. Drouzy, La structuration des ensembles de semigroupes d'ordre 2, 3 et 4 par la relation d'interassociativité, Manuscript, 1986

[4] B. N. Givens, K. Linton, A. Rosin, L. Dishman, “Interassociates of the free commutative semigroup on $n$ generators”, Semigroup Forum, 74 (2007), 370–378 | DOI | MR | Zbl

[5] B. N. Givens, A. Rosin, K. Linton, “Interassociates of the bicyclic semigroup”, Semigroup Forum, 94 (2017), 104–122 | DOI | MR | Zbl

[6] A. B. Gorbatkov, “Interassociates of a free semigroup on two generators”, Mat. Stud., 41 (2014), 139–145 | MR | Zbl

[7] A. B. Gorbatkov, “Interassociates of the free commutative semigroup”, Sib. Math. J., 54:3 (2013), 441–445 | DOI | MR | Zbl

[8] M. Gould, K. A. Linton, A. W. Nelson, “Interassociates of monogenic semigroups”, Semigroup Forum, 68 (2004), 186–201 | DOI | MR | Zbl

[9] M. Gould, R. E. Richardson, “Translational hulls of polynomially related semigroups”, Czechoslovak Math. J., 33:1 (1983), 95–100 | MR | Zbl

[10] J.-L. Loday, “Dialgebras”, Dialgebras and related operads, Lect. Notes Math., 1763, Springer-Verlag, Berlin, 2001, 7–66 | DOI | MR | Zbl

[11] J.-L. Loday, M. O. Ronco, “Trialgebras and families of polytopes”, Contemp. Math., 346 (2004), 369–398 | DOI | MR | Zbl

[12] Yu. M. Movsisyan, Introduction to the Theory of Algebras with Hyperidentities, Yerevan State University Press, Yerevan, 1986 (Russian) | MR

[13] Yu. M. Movsisyan, “Hyperidentitties in algebras and varieties”, Russ. Math. Surveys, 53:1 (1998), 57–108 | DOI | MR | Zbl

[14] T. Pirashvili, “Sets with two associative operations”, Cent. Eur. J. Math., 2 (2003), 169–183 | DOI | MR | Zbl

[15] B. Richter, Dialgebren, Doppelalgebren und ihre Homologie, Diplomarbeit, Universitat Bonn, 1997 ; Available at https://www.math.uni-hamburg.de/home/richter/publications.html | Zbl

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

[17] B. M. Schein, “Restrictive bisemigroups”, Izv. Vyssh. Uchebn. Zaved. Mat., 1:44 (1965), 168–179 (Russian) | Zbl

[18] A. V. Zhuchok, “Commutative dimonoids”, Algebra Discrete Math., 2 (2009), 116–127 | MR | Zbl

[19] A. V. Zhuchok, “Dimonoids and bar-units”, Sib. Math. J., 56:5 (2015), 827–840 | DOI | MR | Zbl

[20] A. V. Zhuchok, Elements of dimonoid theory, Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications, 98, Kiev, 2014, 304 pp. (Ukrainian) | Zbl

[21] A. V. Zhuchok, “Free left $n$-dinilpotent doppelsemigroups”, Commun. Algebra, 45:11 (2017), 4960–4970 | DOI | MR | Zbl

[22] A. V. Zhuchok, M. Demko, “Free $n$-dinilpotent doppelsemigroups”, Algebra Discrete Math., 22:2 (2016), 304–316 | MR | Zbl

[23] A. V. Zhuchok, “Free products of doppelsemigroups”, Algebra Univers., 77:3 (2017), 361–374 | DOI | MR | Zbl

[24] A. V. Zhuchok, “Free rectangular dibands and free dimonoids”, Algebra Discrete Math., 11:2 (2011), 92–111 | MR | Zbl

[25] A. V. Zhuchok, Relatively free doppelsemigroups, Monograph series Lectures in Pure and Applied Mathematics, 5, Potsdam University Press, Germany, Potsdam, 2018, 86 pp. | MR | Zbl

[26] A. V. Zhuchok, “Structure of free strong doppelsemigroups”, Commun. Algebra., 46:8 (2018), 3262–3279 | DOI | MR | Zbl

[27] A. V. Zhuchok, “Structure of relatively free dimonoids”, Commun. Algebra, 45:4 (2017), 1639–1656 | DOI | MR | Zbl

[28] A. V. Zhuchok, “Trioids”, Asian-European J. Math., 8:4 (2015), 1550089, 23 pp. | DOI | MR | Zbl

[29] Yul. V. Zhuchok, “Free rectangular tribands”, Buletinul Academiei de Stiinte a Republicii Moldova. Matematica, 78:2 (2015), 61–73 | MR | Zbl

[30] Y. V. Zhuchok, “Free abelian dimonoids”, Algebra Discrete Math., 20:2 (2015), 330–342 | MR | Zbl

[31] D. Zupnik, “On interassociativity and related questions”, Aequationes Math., 6:2 (1971), 141–148 | DOI | MR | Zbl