I'm reading some books on lattices. Well, I met the notions of compact elements of a lattice and algebraic lattices. The former are the elements $c$ such that for every directed subset D of L, if D has a supremum $\sup D$ and $c \ge \sup D$ then $c \ge d$ for some element $d$ of $D$; while the latter are the lattices in which every element is the supremum of the compact elements below. Now, if we consider a finite lattice, then it is algebraic. What about the infinite case? Can you also suggest me some books and papers investigating the topic?
When an infinite lattices is algebraic?
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$\begingroup$
combinatorics
order-theory
lattice-orders
2 Answers
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Let me modify your definition slightly. Algebraic lattices are: the complete lattices in which every element is the supremum of the compact elements below. The set of integers under the usual order have the property that every element is the supremum of the compact elements below, but $\mathbb Z$ is not algebraic.
The natural examples of algebraic lattices are the lattices of closed subsets of an algebraic closure operator, e.g. any subalgebra lattice of an algebraic structure. A simple example of an infinite lattice that is not algebraic is the unit interval $[0,1]$ of the real line under the usual order.
For a short introduction to algebraic lattices, try JB Nation's notes.
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There are also infinite algebraic lattices. For example, the free lattice generated by a set of chains, $FL(2+2)$, defined here, is infinite, finitely-generated, bounded and algebraic.