Weakly primary semi-ideals in posets

Document Type : Research Paper


Department of Mathematics, Karunya Institute of Technology and Sciences Coimbatore - 641 114, India.



One of the main goals of science and engineering is to avail human beings cull the maximum propitious decisions. To make these decisions, we need to ken human being's predictions, feasible outcomes of various decisions, and since information is never absolutely precise and accurate, we need to withal information about the degree of certainty. All these types of information will lead to partial orders. A partially ordered set (or poset) theory deals with partial orders and plays a major role in real life. It has wide range of applications in various disciplines such as computer science, engineering, medical field, science, modeling spatial relationship in geographic information systems (GIS), physics and so on. In this paper, we mainly focus on weakly primary semi-ideal of a poset. We introduce the concepts of weakly primary semi-ideal and weakly $Q$-primary semi-ideal for some prime $Q$ of a poset $P$ and characterize weakly primary semi-ideals of $P$ in terms of prime and primary semi-ideals of $P.$ We provide a counter-example for the existence of weakly primary semi-ideal of $P$ which is not a primary semi-ideal of $P.$ We found an equivalent assertion of primary (respy., weakly primary) semi-ideal $r(K)$ for a semi-ideal $K$ of $P.$ Moreover, we introduce the notion of direct product of weakly primary semi-ideal of $P$ and describe its characteristics.


[1] A. Badawi, U. Tekir and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc., 51 No. 4 (2014) 1163-1173.
[2] J. Catherine Grace John and B. Elavarasan, Weakly n-prime ideal of posets, Int. J. Pure Appl. Math., 86 No. 6 (2013) 905-910.
[3] S. Ebrahimi Atani and M. Shajari, The diameter of a zero-divisor graph for finite direct product of commutative rings, Sarajevo J. Math., 3 No. 16 (2007) 149-156.
[4] B. Elavarasan and K. Porselvi, An ideal - based zero-divisor graph of posets, Commun. Korean Math. Soc., 28 No. 1 (2013) 79-85.
[5] V. Joshi and N. Mundlik, On primary ideals in posets, Math. Slovaca, 65 No. 6 (2015) 1237-1250.
[6] K. Porselvi and B. Elavarasan, Diameters of semi-ideal based zero-divisor graphs for finite direct product of posets, Ital. J. Pure Appl. Math., 34 (2015) 113-122.
[7] K. Porselvi and B. Elavarasan, Some properties of prime and z-semi-ideals in posets, Khayyam J. Math., 6 No. 1 (2020) 46-56.
[8] R. Hala_s, On extensions of ideals in posets, Discrete Math., 308 (2008) 4972-4977.
[9] P. V. Venkatanarasimhan, Semi-ideals in posets, Math. Ann., 185 No. 4 (1970) 338-348.