Compact lattice element

2020 Mathematics Subject Classification: Primary: 06B23 [MSN][ZBL]

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

[1]	B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1