Franklin's Notes


Given a partial ordered set $P$ with elements $x,y,z\in P$:

If $P$ is such that every pair of elements $x,y\in P$ has both a least upper bound and a greatest lower bound, then we say that $P$ is a lattice. In this case, the join of $x$ and $y$ is denoted $x\lor y$ and their meet is denoted $x\land y$. It can be shown easily that both of these operations are necessarily both commutative and associative. An element $z$ is called join-reducible if it can be written in the form $z=x\lor y$ for some $x,y < z$, and join-irreducible otherwise. It is conventional for the least element of a lattice, if one exists, not to be considered join-irreducible, even though it meets this criterion.

Categorically speaking, a lattice can be thought of as a abstract partial order category with both coproducts and products.




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