(3.6)--3.8-SoftAlgebrasandOptimalSoftAl.pdf

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1、3.83.8 Soft AlgebraSoft Algebras s&Optimal Soft AlgebrasOptimal Soft Algebras Soft algebraSoft algebras s A A soft algebra soft algebra is a is a dual lattice dual lattice with with boundedness boundedness and and distributivitydistributivity.=+.CompletenessCompleteness Let Let(,)be a lattice.If eac

2、h nonempty be a lattice.If each nonempty subset of subset of has both has both a a supremum supremum and an and an infimuminfimum,then,then(,)is called a is called a completecomplete lattice.lattice.Theorem Theorem 1 1 If If(,)is a complete lattice,then it is bounded.is a complete lattice,then it is

3、 bounded.Proof Proof Because Because ,we writewe write =,=.,.(,)is bounded.is bounded.Theorem Theorem 2 2 Let Let,be a complete lattice.Suppose be a complete lattice.Suppose,are nonempty subsets of are nonempty subsets of,and,and ,thenthen ,wherewhere =,=.Prove that if Prove that if ,then,then .Proo

4、f Proof Suppose Suppose .,.Let Let,be a complete lattice.Define be a complete lattice.Define =_,=_.HintHint:Suppose Suppose ,=.We should haveWe should have =,.=.Hint:Hint:Suppose Suppose ,=.We We should should havehave =,.=.Theorem Theorem 3 3 Let Let,be a complete lattice.be a complete lattice.Let

5、Let,.Then Then =(),=().Prove that Prove that =().ProofProof ,().Prove that Prove that =().suppose suppose .or or or or ,.Theorem 4Theorem 4 Let Let,be a complete latticebe a complete lattice.Let.Let be the be the identifier set,identifier set,.Then Then =,=.Generalized Generalized distributivitydist

6、ributivity A A complete lattice is called complete lattice is called generalized generalized distributivedistributive if if =,=hold,wherehold,where ,is the identifier setis the identifier set,.DensityDensity A A lattice lattice,is called is called densedense if for all if for all,in in,if if ,i.e.,i

7、.e.,andand ,then there then there exists an element exists an element in in,such that,such that .Optimal Optimal soft soft algebrasalgebras An An optimal soft algebra optimal soft algebra is a dual lattice which is a dual lattice which is is dense and generalized distributive.dense and generalized distributive.=+.=+.=+.=+._._._.=+.=+.=+.