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If you have a handle on the join operator, then you can try to show that your poset is a complete join semi-lattice. This shows that your poset is also a complete meet semi-lattice, and hence a complete lattice.

This is certainly way more useful in the infinite setting, but the point is you only have to do half the work. For example let $L$ be the poset of subspaces of a vector space $V$ (ordered by inclusion). It is almost trivial that the intersection of an arbitrary collection of subspaces is a subspace. So, $L$ is a complete meet semi-lattice and hence a lattice.

If you have a handle on the join operator, then you can try to show that your poset is a complete join semi-lattice. This shows that your poset is also a complete meet semi-lattice, and hence a complete lattice.

This is certainly way more useful in the infinite setting, but the point is you only have to do half the work. For example let $L$ be the poset of subspaces of a vector space $V$ (ordered by inclusion). It is almost trivial that the intersection of an arbitrary collection of subspaces is a subspace. So, $L$ is a complete meet semi-lattice and hence a lattice.

If you have a handle on the join operator, then you can try to show that your poset is a complete join semi-lattice. This shows that your poset is also a complete join semi-lattice, and hence a complete lattice.

This is certainly way more useful in the infinite setting, but the point is you only have to do half the work. For example let $L$ be the poset of subspaces of a vector space $V$ (ordered by inclusion). It is almost trivial that the intersection of an arbitrary collection of subspaces is a subspace. So, $L$ is a complete meet semi-lattice and hence a lattice.

If you have a handle on the join operator, then you can try to show that your poset is a complete join semi-lattice. This shows that your poset is also a complete meet semi-lattice, and hence a complete lattice.

This is certainly way more useful in the infinite setting, but the point is you only have to do half the work. For example let $L$ be the poset of subspaces of a vector space $V$ (ordered by inclusion). It is almost trivial that the intersection of an arbitrary collection of subspaces is a subspace. So, $L$ is a complete meet semi-lattice and hence a lattice.

If you have a handle on the join operator, then you can try to show that your poset is a complete join semi-lattice. This shows that your poset is also a complete meet semi-lattice, and hence a complete lattice.

This is certainly way more useful in the infinite setting, but the point is you only have to do half the work. For example let $L$ be the poset of subspaces of a vector space $V$ (ordered by inclusion). It is almost trivial that the intersection of an arbitrary collection of subspaces is a subspace. So, $L$ is a complete meet semi-lattice and hence a lattice.

If you have a handle on the join operator, then you can try to show that your poset is a complete join semi-lattice. This shows that your poset is also a complete join semi-lattice, and hence a complete lattice.

This is certainly way more useful in the infinite setting, but the point is you only have to do half the work. For example let $L$ be the poset of subspaces of a vector space $V$ (ordered by inclusion). It is almost trivial that the intersection of an arbitrary collection of subspaces is a subspace. So, $L$ is a complete meet semi-lattice and hence a lattice.