Some properties of 1-distributive join semilattices

admin May 17, 2015

Authors:

Shiuly Akhter,A.S.A.Noor,M.Ayub Ali,

DOI NO:

https://doi.org/10.26782/jmcms.2015.01.00002

Keywords:

1-distributive joinsemi lattice ,Semi prime filter,Prime ideal,Maximal ideal,-filter,

Abstract

J.C.Varlet introduced the concept of 1-distributive lattices to generalize thenotion ofdualpseudo complemented lattices. A latticeLwith 1 is calleda 1-distributivelattice if for allLc,b,a,caba1imply1)(cba. Of course everydistributive lattice with 1 is 1-distributive. Also everydual pseudo complemented lattice is1-distributive.Recently, Shiuly and Noor extended this concept for directedbelow joinsemi lattices. A joinsemi latticeSis calleddirected belowif for allSb,a, there existsScsuch thatb,ac. Again Y.Rav has extended the concept of 1-distributivity byintroducing the notion ofsemi prime filtersin a lattice. Recently, Noor and Ayubhavestudied the semi prime filters in a directed below joinsemi lattice. In this paper we haveincluded several characterizations and properties of 1-distributive joinsemi lattices.Weproved that for a joinsub semi latticeAofS,Aasomeforax:SxA11is a semi prime filter ofSif and only ifSis 1-distributive.We also showed that a directed below join semi lattice with 1 is 1-distributiveif and only if for allSb,a,111)()()(dbafor someb,ad,Sd.Introducingthe notion of-filters and using different equivalent conditions of 1-distributive joinsemilattices we have given a ‘Separation theorem’ for-filters.

Refference:

I.M. Ayub Ali and A. S. A. Noor, Semiprime filters in join semilattices,Submitted in Annals of Pure & Applied Mathematics,India.

II.P. Balasubramani and P. V. Venkatanarasimhan,Characterizations of the 0-Distributive Lattices,Indian J. Pure appl.Math. 32(3) 315-324, (2001).

III.R. M. Hafizur Rahman, M. Ayub Ali and A. S. A. Noor, On Semiprime ideals oflattices, J.Mech. Cont. & Math. Sci. Vol.-7, No.-2, January (2013), pages 1094-1102.

IV.C. Jayaram,0-modular semilattices, Studia Sci. Math. Hung. 22(1987), 189-195.

V.Y. S. Powar and N. K. Thakare,0-Distributive semilattices, Canad. Math. Bull.Vol. 21(4) (1978), 469-475.

VI.Y. Rav,Semi prime ideals in general lattices,Journal of pure and AppliedAlgebra, 56(1989) 105-118.

VII.Shiuly Akhter and A.S.A.Noor,1-distributive join semilattice, J. Mech.Cont. & Math. Sci. Vol.-7,No.-2,January (2013) pages 1067-1076.

VIII.J. C. Varlet,A generalization of the notion of pseudo-complementedness,Bull.Soc.Sci.Liege, 37(1968), 149-158.

Author(s) : Shiuly Akhter, A. S. A. Noor, and M. Ayub Ali View Download