《(3.3)--3.5-LatticesPart2.pdf》由会员分享，可在线阅读，更多相关《(3.3)--3.5-LatticesPart2.pdf（16页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、3.53.5 Lattices:Part 2Lattices:Part 2 Lemma 1Lemma 1 Let Let(,)be the algebraic structure induced be the algebraic structure induced by by the lattice the lattice,.If.If,withwith ,thenthen ,.EspeciallyEspecially,if,if ,thenthen ,.Prove that if Prove that if ,thenthen .ProofProof ,=,.,.Theorem Theore
2、m 1 1 Let Let(,)be the algebraic structure induced be the algebraic structure induced by the lattice by the lattice(,).If.If,，then then =.Prove that Prove that =.Proof Proof Suppose Suppose .=,=.Suppose Suppose =.=,=.Theorem 2 Theorem 2 Let Let(,)be the algebraic be the algebraic structure induced b
3、y the lattice structure induced by the lattice(,).Then.Then the two binary operations the two binary operations and and satisfy the satisfy the following properties:following properties:=,=.=,=.Prove that Prove that =.Proof Proof =,=,=()=.Lemma 2Lemma 2 Let Let(,)be an algebraic structure.If the be
4、an algebraic structure.If the absorption laws hold true,then the idempotent absorption laws hold true,then the idempotent laws hold truelaws hold true.Prove Prove thatthat =.Proof Proof Suppose Suppose =,=.=()=.Suppose Suppose the the algebraic structure algebraic structure(,)is is induced by the la
5、ttice induced by the lattice(,),：,=,.：,=,.Theorem Theorem 3 3 Let Let(,)be an algebraic structure with two be an algebraic structure with two binary operations which obey binary operations which obey commutative commutative laws,laws,associativeassociative laws laws and and absorptionabsorption laws
6、.laws.Then there exists a Then there exists a lattice lattice(,)such that the such that the algebraic structure induced by this lattice is algebraic structure induced by this lattice is(,).Analysis Analysis Suppose Suppose(,)is a is a latticelattice,(,)is the is the algebraic structure induced by al
7、gebraic structure induced by(,),then then =.Proof Proof Define Define =,.(,)is is a a posetposet.(,)is a is a latticelattice.,=,=.Let L be a lattice.DefineLet L be a lattice.Define ：,=,.：，,=,.SupposeSuppose (,)is the is the algebraic structure algebraic structure induced by the lattice induced by the lattice(,).(,)induced by lattice induced by lattice(,).=,=.=,=.=,=.=,=.=.Algebraic Algebraic structurestructure (,).=,=.=,=.=,=.DefineDefine =,=,=.=,.latticelattice(,),=,=.lattice lattice,(,)=,=,.Two Two equivalent equivalent definitions of latticesdefinitions of lattices