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.
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 ''P'', if <math>S\subseteq I</math> then <math>\operatorname{LU}(S)\subseteq I</math>.
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 ''P'', if ''S'' <math>\subseteq</math> ''I'' then LU(''S'') <math>\subseteq</math> ''I''.
== Remarks== #Every Frink ideal ''I'' is a Doyle pseudoideal. #A subset ''I'' of a lattice (''P'', ≤) is a Doyle pseudoideal if and only if it is a lower set that is closed under finite joins (suprema).
==Related notions== *Frink ideal
==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, 56: 366. *Niederle, J. (2006) "Ideals in ordered sets", Rendiconti del Circolo Matematico di Palermo 55: 287–295.
Category:Order theory