Lattice defined on poset vs. Lattice defined on group?

Question

I've seen two different definitions of the term lattice, one is defined on poset, the other one is defined on group. I believe these two are fundamentally different mathematical objects. But I'm not a hundred percent sure that these two has nothing to do with each other since the lattice defined on poset can alternatively be defined as algegraic structures. Can anyone clarify it?

Answer 1

As far as I’m aware, there’s no formal connection, but here are some points where a connection can be drawn and where you can see how people might have come up with the idea of calling order-theoretic lattices “lattices”:

• The examples in the Wikipedia article on order-theoretic lattices – a lot of them look a lot like lattices.
• The game of Chomp, where you select which parts of a chocolate lattice to eat by selecting a lattice point and eating everything “less than” that point. On such a lattice (in both senses of the word), the join and meet are simply defined as the coordinatewise maximum and minimum, respectively.
• That’s an example of a product order – and if you look at that article, again the illustration looks a lot like a lattice.

This has nothing to do with the fact that lattices in the order-theoretic sense can alternatively be defined as algebraic structures. Those algebraic structures don’t necessarily have any of the properties that lattices in the group-theoretic sense have – they need not be groups, they need not be infinite, they need not be regular. The algebraic view of lattices is merely a reformulation (though a useful one) of the order-theoretic view.