Some properties of 1-distributive join semilattices


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



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


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.


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