View source for Compact lattice element

{{TEX|done}}{{MSC|06B23}}

An element $a$ of a [[complete lattice]] $L$ for which the condition
$$
a \le \bigvee_{j \in J} x_j\,,\ \ x_j \in L\,,
$$
implies
$$
a \le x_{j_1} \vee \cdots \vee x_{j_k}
$$
for some finite subset $\{j_1,\ldots,j_k\} \subset J$.

An [[algebraic lattice]] is one in which each element is the union (least upper bound) of a set of compact elements.  

A '''finite''' element $b$ of a lattice $L$ is one for which the condition
$$
b \le \bigvee_{d \in D} d
$$
for a [[directed set]] $D \subset L$ implies
$$
b \le d
$$
for some $d \in D$.

In a complete lattice, the compact elements are precisely the finite elements.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1</TD></TR>
</table>