# Product lattice

Could someone explain how to construct a product lattice, or point me to an explanation on the web?

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I recommend that you read Davey and Priestley "Introduction to Lattices and Order". Specifically sections 1.26 and 2.14. –  Loop Space Jun 1 '10 at 12:42
I don't have a copy! :( And I need to multiply two lattices... –  user6473 Jun 1 '10 at 12:46

Please click through and read the discussion there. But to summarize, one choice is to form the product order, where you consider pairs $(a,b)$ and the orders are inherited in each coordinate separately. Another choice is to use the lexical order on pairs, but this only results in a lattice when there are certain completeness or linearity assumptions. The lexical product $K\times L$, however, amounts to replacing each node in $K$ with a copy of $L$, and this is often what is desired.