Is there a connection between lattices in the sense of orders and lattices in the sense of groups?

Question

I'm wondering about this for some time now - is there a intuitive connection between those concepts or have they been named the same by chance?

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In particular, my interest in this question was sparked by the fact that $\mathbb{Z}^d$ is an integer lattice / vector lattice / lattice in the group sense, but $(\mathbb{Z}^d,\leq )$ is also a lattice in the order sense.

So I'm, wondering if maybe historically, people considered objects that are lattices in both senses up to a point where they split the notation and two different notions of lattice came to be. The other version would be that people just named the same way independently.

I'm also wondering if one of the two concepts can always be considered as a special case of the other one, things like that. I would guess every lattice in the group sense can inherit a partial order from $(\mathbb{Z}^d,\leq )$ via its parametrization or something like that.

Answer 1

Following up on what Qiaochu Yuan said:

This is a picture of what google says a real life lattice is:

Here is the lattice of a group (partial order)

Here is a plane lattice