In the theory of
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 ...
s, a pseudoideal is a subset characterized by a bounding operator LU.
Basic definitions
LU(''A'') is the set of all
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 element ...
s of the set of all
upper bounds 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 Doyle pseudoideal, if the following condition holds:
For every finite subset ''S'' of ''P'' that has a
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
in ''P'', if
then
.
A subset ''I'' of a partially ordered set (''P'', ≤) is a pseudoideal, if the following condition holds:
For every subset ''S'' of ''P'' having at most two elements that has a
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
in ''P'', if ''S''
''I'' then LU(''S'')
''I''.
Remarks
#Every
Frink ideal In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set.
Basic definitions
LU(''A'') is the set of all common lower bounds of the set of all common upper bounds of the subset ''A'' of a par ...
''I'' is a Doyle pseudoideal.
#A subset ''I'' of a lattice (''P'', ≤) is a Doyle pseudoideal
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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
).
Related notions
*
Frink ideal In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set.
Basic definitions
LU(''A'') is the set of all common lower bounds of the set of all common upper bounds of the subset ''A'' of a par ...
References
*Abian, A., Amin, W. A. (1990) "Existence of prime ideals and ultrafilters in partially ordered sets", Czechoslovak Math. J., 40: 159–163.
*Doyle, W.(1950) "An arithmetical theorem for partially ordered sets",
Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...
, 56: 366.
*Niederle, J. (2006) "Ideals in ordered sets",
Rendiconti del Circolo Matematico di Palermo 55: 287–295.
Order theory