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A finite lattice $L$ is called modular if and only if its elements satisfy the following modular identity: For all $x,y,z\in L$ such that $x\leq z$, we have $x\vee(y\wedge z)=(x\vee y)\wedge z$.

How can we define an infinite modular lattice? Does the criterion remain the same in the infinite case as well?

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closed as off-topic by user21820, Adrian Keister, The Count, postmortes, GNUSupporter 8964民主女神 地下教會 Jun 25 at 22:28

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Of course, it's a condition that has to be checked for all triples $x,y,z$ with $x \le z$, but it's a simple "algebraic" condition between a few elements. No distinction is needed (or made) for the case where $L$ itself is infinite or not.

The condition is what makes a lattic modular, not its size. Maybe your text has some special theorems that only hold for finite lattices, and wants to apply those? There is no reason to restrict the size in the definition of modular.

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