Abstract
The semilattice congruence \(\mathscr {N}\), which identifies two elements if they generate the same principal filter, plays a significant role in studying the decomposition of semigroups. We investigate the remarkable properties of the semilattice congruence \(\mathscr {N}\) on n-ary semigroups, where \(n\ge 3\), and use these properties to describe the structure of n-ary semigroups which are decomposable into i-simple and regular components for all \(1<i<n\). In particular, we show that each n-ary semigroup which is both regular and intra-regular is decomposable into a semilattice of i-simple and regular n-ary semigroups, and the reverse assertion also holds. Moreover, we prove that an n-ary semigroup is intra-regular if and only if it is a semilattice of i-simple n-ary semigroups. Finally, we discuss the connection between semilattices of i-simple (and regular) n-ary semigroups and semilattices of simple (and regular) n-ary semigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Basar, A., Abbasi, M.Y.: On generalized bi-\(\Gamma \)-ideals in \(\Gamma \)-semigroups. Quasigroups Relat. Syst. 23, 181–186 (2015)
Bogdanović, S.M., Ćirić, M.D., Popović, Z.L.: Semilattice Decompositions of Semigroups. University of Niš, Faculty of Economics, Niš (2011)
Clifford, A.H.: Semigroups admiting relative inverse. Ann. Math. 42, 1037–1042 (1941)
Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Vol. I, Mathematical Surveys No. 7. American Mathematical Society, Providence (1961)
Devillet, J., Mathonet, P.: On the structure of symmetric \(n\)-ary bands. Int. J. Algebra Comput. 37, 1319–1338 (2021)
Dixit, V.N., Dewan, S.: A note on quasi and bi-ideals in ternary semigroups. Int. J. Math. Math. Sci. 18, 501–508 (1995)
Dudek, W.A.: On divisibility in \(n\)-semigroups. Demonstr. Math. 13, 355–367 (1980)
Dudek, W.A.: On \((i, j)\)-associative \(n\)-groupoids with the non-empty center. Ric. Mat. 35, 105–111 (1986)
Dudek, W.A.: Idempotents in \(n\)-ary semigroups. Southeast Asian Bull. Math. 25, 97–104 (2001)
Dudek, W.A., Groździńska, I.: On ideals in regular \(n\)-semigroups. Mat. Bilten (Skopje) 3-4, 35–44 (1979-1980)
Grzymala-Busse, J.W.: Automorphisms of polyadic automata. J. Assoc. Comput. Mach. 16, 208–219 (1969)
Kasner, E.: An extension of the group concept (reported by L.G. Weld). Bull. Am. Math. Soc. 10, 290–291 (1904)
Kehayopulu, N., Tsingelis, M.: On ordered semigroups which are semilattices of simple and regular semigroups. Comm. Algebra 41, 3252–3260 (2013)
Kehayopulu, N., Tsingelis, M.: On intra-regular and some left regular \(\Gamma \)-semigroups. Quasigroups Relat. Syst. 23, 263–270 (2015)
Lajos, S.: On generalized bi-ideals in semigroups. Coll. Math. Soc. János Bolyai 20, Algebraic Theory of Semigroups, (G. Pollák, Ed.), pp. 335–340. North–Holland, Amsterdam (1979)
Mitrović, M.S.: Semilattices of Archimedean Semigroups. University of Niš, Faculty of Mechanical Engineering, Niš (2003)
Nikshych, D., Vainerman, L.: Finite quantum groupoids and their applications. ArXiv preprint (2000) https://doi.org/10.48550/arxiv.math/0006057
Petchkaew, P., Chinram, R.: The minimality and maximality of \(n\)-ideals in \(n\)-ary semigroups. Eur. J. Pure Appl. Math. 11(3), 762–773 (2018)
Petrich, M.: Introduction to Semigroups. Merrill Publishing Co., Columbus (1973)
Pop, M.S.: On simple \(n\)-semigroups. Bul. Ştiinţ. Univ. Baia Mare Ser. B. 15, 33–44 (1999)
Pornsurat, P., Ayutthaya, P.P., Pibaljommee, B.: Prime \(i\)-ideals in ordered \(n\)-ary semigroups. Mathematics 9, 491 (2021)
Preston, G.B.: The arithmetic of a lattice of sub-algebras of a general algebra. J. Lond. Math. Soc. 29, 1–15 (1954)
Sioson, F.M.: On regular algebraic systems. Proc. Jpn. Acad. 39, 283–286 (1963)
Somsup, C., Leerawat, U.: Congruences and homomorphisms on \(n\)-ary semigroups. Int. J. Math. Comp. Sci. 15, 671–682 (2020)
Stojaković, Z., Dudek, W.A.: Single identities for varieties equivalent to quadruple systems. Discrete Math. 183, 277–284 (1998)
Tamura, T.: Another proof of a theorem concerning the greatest semilattice decomposition of a semigroup. Proc. Jpn. Acad. 40, 777–780 (1964)
Tamura, T.: Note on the greatest semilattice decomposition of semigroups. Semigroup Forum 4, 255–261 (1972)
Tamura, T., Kimura, N.: On decomposition of a commutative semigroup. Kodai Math. Sem. Rep. 4, 109–112 (1954)
Tamura, T., Kimura, N.: Existence of greatest decomposition of a semigroup. Kodai Math. Sem. Rep. 7, 83–84 (1955)
Yamada, M.: On the greatest semilattice decomposition of a semigroup. Kodai Math. Sem. Rep. 7, 59–62 (1955)
Acknowledgements
The authors are highly grateful to the referees for their valuable comments and suggestions for improving the article. This research was supported by Chiang Mai University, Chiang Mai 50200, Thailand.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jorge Almeida.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Daengsaen, J., Leeratanavalee, S. Semilattices of simple and regular n-ary semigroups. Semigroup Forum 107, 294–314 (2023). https://doi.org/10.1007/s00233-023-10375-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-023-10375-w