In mathematics, a Frink ideal, introduced by
Orrin Frink
Orrin Frink Jr. (31 May 1901 – 4 March 1988). was an American mathematician who introduced Frink ideals in 1954.
Frink earned a doctorate from Columbia University in 1926 or 1927 and worked on the faculty of Pennsylvania State University for 41 ...
, is a certain kind of subset of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
.
Basic definitions
LU(''A'') is the set of all common
lower bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
s of the set of all common
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
s of the subset ''A'' of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
.
A subset ''I'' of a partially ordered set (''P'', ≤) is a Frink ideal, if the following condition holds:
For every finite subset ''S'' of ''I'', we have LU(''S'')
''I''.
A subset ''I'' of a partially ordered set (''P'', ≤) is a normal ideal or a cut if LU(''I'')
''I''.
Remarks
#Every Frink ideal ''I'' is a
lower set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
.
#A subset ''I'' of a lattice (''P'', ≤) is a Frink ideal
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
it is a lower set that is closed under finite joins (
suprema).
# Every normal ideal is a Frink ideal.
Related notions
*
pseudoideal In the theory of partially ordered sets, a pseudoideal is a subset characterized by a bounding operator LU. Basic definitions
LU(''A'') is the set of all lower bounds of the set of all upper bounds of the subset ''A'' of a partially ordered set.
...
*
Doyle pseudoideal
References
*
*{{cite journal, author=Niederle, Josef, title=Ideals in ordered sets, journal=
, volume=55, year=2006, pages=287–295, doi=10.1007/bf02874708, s2cid=121956714
Order theory