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2004•Neimenggu Shi-da xuebao. Zhexue shehui kexue hanwen banRequires access

THE NECESSARY AND SUFFICIENT CONDITIONS OF THE COMPLETE RINGS OF SETS BEING THE IDEAL LATTICES OF SEMILATTICES OR LATTICES

Yuqi Zhang

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Abstract

In this paper,The results below are proved:A complete lattice L is a complete ring of sets  (LI(F),I(F)) is the lattice of all ideals of F,and F is a distributive join semilattice which is finitely generated by the set of all completely join irreducible elements of L.If a complete lattice L is isomorphic to the ideal lattice of a lattice K,i.e.,I(K)L,then L is a complete ring of sets  K is a strong Sober lattice.

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What this paper is about

In this paper,The results below are proved:A complete lattice L is a complete ring of sets  (LI(F),I(F)) is the lattice of all ideals of F,and F is a distributive join semilattice which is finitely generated by the set of all completely join irreducible elements of L.If a complete lattice L is isomorphic to the ideal lattice of a lattice K,i.e.,I(K)L,then L is a complete ring of sets  K is a strong Sober lattice.

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Available abstract

In this paper,The results below are proved:A complete lattice L is a complete ring of sets  (LI(F),I(F)) is the lattice of all ideals of F,and F is a distributive join semilattice which is finitely generated by the set of all completely join irreducible elements of L.If a complete lattice L is isomorphic to the ideal lattice of a lattice K,i.e.,I(K)L,then L is a complete ring of sets  K is a strong Sober lattice.

Key concepts: Semilattice, Distributive lattice, Lattice (music), Mathematics, Join (topology), Congruence lattice problem, Complete lattice, Distributive property

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