Abstract
Here we show that every normal band N can be embedded into the normal band \(\mathcal {B(S)}\) of all k-bi-ideals, the left part \(N/ \mathcal {R}\) of N into the left normal band \(\mathcal {R(S)}\) of all right k-ideals, the right part \(N/ \mathcal {L}\) of N into the right normal band \(\mathcal {L(S)}\) of all left k-ideals, and the greatest semilattice homomorphic image \(N/ \mathcal {J}\) of N into the semilattice of all k-ideals of a same k-regular and intra k-regular semiring S.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adhikari, M.R., Sen, M.K., Weinert, H.J.: On \(k\)-regular semirings. Bull. Cal. Math. Soc. 88, 141–144 (1996)
Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra. In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds.) Handbook of Linear Algebra. Chapman & Hall, London (2006)
Berstel, J.: Rational Series and Their Languages. Monogr. Theoret. Comput. Sci. EATCS Ser., vol. 12. Springer, Berlin (1988)
Bhuniya, A.K., Jana, K.: Bi-ideals in \(k\)-regular and intra \(k\)-regular semirings. Discuss. Math. Gen. Algebra Appl. 31, 5–23 (2011)
Bourne, S.: The Jacobson radical of a semiring. Proc. Natl. Acad. Sci. USA 37, 163–170 (1951)
Conway, J.H.: Regular Algebra and Finite Machines. Chapman & Hall, London (1971)
Develin, M., Santos, F., Sturmfels, B.: On the Rank of a tropical matrix. In: Combinatorial and Computational Geometry. MSRI Publications, vol. 52, pp. 213–242. Cambridge University Press, Cambridge (2005)
Gathmann, A.: Tropical algebraic geometry. Jahresber. Deutsch. Math. Verein. 108, 3–32 (2006)
Good, R.H., Hughes, D.R.: Associated groups for semigroup. Bull. Am. Math. Soc. 58, 624–625 (1952)
Hebisch, U., Weinert, H.J.: Semirings: Algebric Theory and Applications in Computer Science. World Scientific, Singapore (1998)
Howie, J.M.: Fundamentals in Semigroup Theory. Clarendon, Oxford (1995)
Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry. Oberwolfach Semin., vol. 35, 2nd edn. Birkhuser, Basel (2009)
Izhakian, Z., Rowen, L.: Supertropical algebra. Adv. Math. 225, 2222–2286 (2010)
Litvinov, G.L., Maslop, V.P.: The correspondence principle for idempotent calculas and some computer applications. In: Gunawardena, J. (ed.) Idempotency, pp. 420–443. Cambridge Univ. Press, Cambridge (1998)
Litvinov, G.L., Maslop, V.P., Shpiz, G.B.: Idempotent functional analysis. Mat. Zametki. 69, 758–797 (2001)
Litvinov, G.L., Maslop, V.P., Sobolevskii, A.N.: Idempotent Mathematics and Interval Analysis. The Erwin Schrödinger International Institute for Mathematical Physics, Vienna (1998)
Nambooripad, K.S.S.: Pseudo-semilattices and biordered sets-1. Simon Stevin 55, 103–110 (1981)
Polák, L.: Syntactic semiring of a language (extended abstract). In: Láznĕ, M. (ed.) Mathematical Foundations of Computer Science. Lecture Notes in Comput. Sci., vol. 2136, pp. 611–620. Springer, Berlin (2001)
Polák, L.: Syntactic semiring and language equations. In: Proceedings of the Seventeenth International Conference on Implementation and Application of Automata. Lecture Notes in Computer Science, vol. 2608, pp. 182–193. Springer, Berlin (2003)
Polák, L.: Syntactic semiring and universal automaton. In: Proceedings of the International Conference on Developments in Language Theory. Lecture Notes in Computer Science, vol. 2710, pp. 411–422. Springer, Berlin (2003)
Polák, L.: A classification of rational languages by semilattice-ordered monoids. Arch. Math. (Brno) 40, 395–406 (2004)
Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Proceedings of the International Workshop on Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics, vol. 377, pp. 289–317. American Mathematical Society, Providence (2005)
Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts Monogr. Comput. Sci. Springer, New York (1978)
Schützenberger, M.P.: On the definition of a family of automata. Inf. Control. 4, 245–270 (1961)
Sen, M.K., Bhuniya, A.K.: On additive idempotent \(k\)-clifford semirings. Southeast Asian Bull. Math. 32, 1149–1159 (2008)
Vandiver, H.S.: Note on a simple type of algebra in which the cancellation law of addition does not hold. Bull. Am. Math. Soc. 40, 914–920 (1934)
Zalcstein, Y.: Locally testable semigroup. Semigroup Forum. 5, 216–227 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by M. Jackson.
This article is dedicated to Prof. M. K. Sen on his 78th birthday.
Rights and permissions
About this article
Cite this article
Bhuniya, A.K., Jana, K. Representation of normal bands as semigroups of k-bi-ideals of a semiring. Algebra Univers. 79, 6 (2018). https://doi.org/10.1007/s00012-018-0481-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-018-0481-4