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I encountered the following statement in a non-math book (on digital systems, actually), where the author discusses lattices and Boolean algebras:

The following properties are valid for every finite lattice:
$a+0=a$ and $a.0=0$

I can't see how the proof of these properties could not be valid for an infinite lattice? Could someone give any counter examples to this?

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    Must infinite lattices have a smallest element? – 2017-01-11
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    To begin with, an infinite lattice may not have a least element. – 2017-01-11
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    Some people (unfortunately) define lattices in a way that doesn't require the existence of a smallest element. In a finite lattice, that would follow from the other requirements, but an infinite lattice could, under such a definition, not have a $0$ element at all. In that case, the equations you're asking about wouldn't make sense. Whenever a $0$ element exists in a lattice, it satisfies these equations. – 2017-01-11
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    @AndreasBlass Why "unfortunately?" That is, why exclude lattices that are not complete? – 2017-01-11
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    @FabioSomenzi I don't want to require latices to be complete; my preferred definition is that they are **finitely** complete, i.e., every finite subset has a greatest lower bound and a least upper bound. In particular, the empty set has these bounds, 1 and 0, respectively. – 2017-01-11
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    Ah, I get it now.. the existence, and therefore uniqueness, of 0 and 1 is dependent on the fact that the lattice is finite – 2017-01-11
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    @AndreasBlass Thanks for the clarification. So, the set of all finite subsets of an infinite set is not finitely complete (under set inclusion) because it doesn't have a greatest element. – 2017-01-11
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    @ArkyaChatterjee Almost right. An infinite lattice may have 0 and 1. Think of the powerset of an infinite set. The empty set is 0 and the whole set is 1. When the lattice is finite, then it *must* have a 0 and a 1. – 2017-01-11
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    @FabioSomenzi Yeah, that's what I meant, that the certainty of existence of 0 and 1 in a finite lattice hinges on its finiteness; and hence will not in general go through for an infinite lattice.. thanks a lot! – 2017-01-11

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To cut down on the number of unanswered questions on this site, I post this answer.

Consider the lattice $\langle\Bbb Z_-,<\rangle,$ where $\Bbb Z_-$ is the set of negative integers and $<$ is the usual order relation on this set. Readily, there is no such thing as a $0$ of this lattice.