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单词 DifferenceOfLatticeElements
释义

difference of lattice elements


Let 𝔄 is a latticeMathworldPlanetmath with least element 0.

Let a,b𝔄. A of a and b is an element c𝔄 that bc=0 and ab=bc. When there is only one differencePlanetmathPlanetmath of a and b, it is denoted ab.

One immediate property is: 0 is the unique difference of any element a and itself (aa=0). For if c is such a difference, then ac=0 and a=ac. So ca by the second equation, and hence that c=ac=0 by the first equation.

For arbitrary lattices of two given elements do not necessarily exist. For some lattices there may be more than one difference of two given elements.

For a distributive latticeMathworldPlanetmath with bottom element 0, the difference of two elements, if it exists, must be unique. To see this, let c and d be two differences of a and b. Then

  • bc=bd=0, and

  • ab=bc=bd.

So c=c(bc)=c(bd)=(cb)(cd)=0(cd)=cd. Similarly, d=dc. As a result, c=cd=dc=d.

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更新时间:2025/5/4 17:31:18