Difference between revisions of "Compact lattice element"
From Encyclopedia of Mathematics
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| − | An element $a$ of a lattice $L$ for which the condition | + | An element $a$ of a [[complete lattice]] $L$ for which the condition |
$$ | $$ | ||
a \le \bigvee_{j \in J} x_j\,,\ \ x_j \in L\,, | a \le \bigvee_{j \in J} x_j\,,\ \ x_j \in L\,, | ||
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$$ | $$ | ||
for some finite subset $\{j_1,\ldots,j_k\} \subset J$. | 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> | ||
Latest revision as of 08:06, 26 November 2023
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 |
How to Cite This Entry:
Compact lattice element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compact_lattice_element&oldid=36183
Compact lattice element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compact_lattice_element&oldid=36183
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article