Characterizations of $J$-prime ideals and $M_{J}$-ideals in posets

Document Type : Research Paper

Authors

Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore-641114, Tamilnadu, India.

10.29252/as.2022.2719

Abstract

In this paper, we introduce the concepts of $J$-prime ideals and $M_{J}$-ideals in posets, and obtain some of their interesting characterizations in posets. Furthermore, we discuss the properties of $J$-ideals that are analogous to $J$-prime ideals and $M_J$-ideals in posets. Finally, we establish a set of equivalent conditions for an ideal in a poset $\mathcal{P}$ containing an ideal $J$ is an $J$-ideal, and for a semi-prime ideal $J$ to be an $M_{J}$-ideal of $\mathcal{P}$.

Keywords


[1] W. H. Cornish, Annulets and α-ideals in a distributive lattice, J. Aust. Math. Soc., 15 No. 1 (1973) 70-77.
[2] R. Halas, On extensions of ideals in posets, Discrete Math., 308 No. 21 (2008) 4972-4977.
[3] C. Jayaram, Prime α-ideals in a 0-distributive lattice, Indian J. pure appl. Math., 17 No. 3 (1986) 331-337.
[4] J. C. G. John and B. Elavarasan, Weakly n-prime ideal of posets, Int. J. Pure Appl. Math., 86 No. 6 (2013) 905-910.
[5] J. C. G. John and B. Elavarasan, zJ-Ideals and Strongly Prime Ideals in Posets, Kyungpook Math. J., 57 No. 3 (2017) 385-391.
[6] J. C. G. John and B. Elavarasan, Properties of M-ideals in partially ordered sets, Adv. Math. Sci. J., 9 No. 2 (2020) 601-606.
[7] J. C. G. John and B. Elavarasan, Properties of J-ideals in posets, Springer book series-Lect. Notes Netw. Syst., (Accepted).
[8] V. Joshi and N. Mundlik, Baer ideals in 0-distributive posets, Asian Eur. J. Math., 9 No. 3 (2016) 1650055.
[9] V. S. Kharat and K. A. Mokbel, Primeness and semiprimeness in posets, Math. Bohem., 134 No. 1 (2009) 19-30.
[10] K. A. Mokbel, α-ideals in 0-distributive posets, Math. Bohem., 140 No. 3 (2015) 319-328.
[11] Y. Pawar and S. Khopade, α-ideals and annihilator ideals in 0-distributive lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 49 No. 1 (2010) 63-74.
[12] P. Venkatanarasimhan, Semi-ideals in posets, Math. Ann., 185 No. 4 (1970) 338-348.