分配格




设(L,∨,∧){displaystyle (L,vee ,wedge )}(L, vee, wedge)是一个格，若对于任意的a,b,c∈L{displaystyle a,b,cin L}a, b, c in L有




a∧(b∨c)=(a∧b)∨(a∧c){displaystyle awedge (bvee c)=(awedge b)vee (awedge c)}{displaystyle awedge (bvee c)=(awedge b)vee (awedge c)}


a∨(b∧c)=(a∨b)∧(a∨c){displaystyle avee (bwedge c)=(avee b)wedge (avee c)}{displaystyle avee (bwedge c)=(avee b)wedge (avee c)}


则称L{displaystyle L}L为分配格。


上述两个等式互为对偶式，根据格的对偶原理，在证明一个格是分配格时只需证明其中任意一个等式即可。


设(L,∨,∧){displaystyle (L,vee ,wedge )}(L, vee, wedge)是一个格，L{displaystyle L}L为分配格当且仅当对于任意的a,b,c∈L{displaystyle a,b,cin L}a, b, c in L，若a∨b=a∨c{displaystyle avee b=avee c}{displaystyle avee b=avee c}且a∧b=a∧c{displaystyle awedge b=awedge c}{displaystyle awedge b=awedge c}，则b=c{displaystyle b=c}b=c。



参见



  • 格

  • 有补格

  • 有界格

  • 布尔代数








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