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order ideal (Definition)

Order Ideals and Filters

Let $ P$ be a poset. A subset $ I$ of $ P$ is said to be an order ideal if

An order ideal is also called an ideal for short. An ideal is said to be principal if it has the form $ \downarrow\!\!x$ for some $ x\in P$.

Given a subset $ A$ of a poset $ P$, we say that $ B$ is the ideal generated by $ A$ if $ B$ is the smallest order ideal (of $ P$) containing $ A$. $ B$ is denoted by $ \langle A\rangle$. Note that $ \langle A\rangle$ exists iff $ A$ is a directed set. In particular, for any $ x\in P$, $ \downarrow\!\!x$ is the ideal generated by $ x$. Also, if $ P$ is an upper semilattice, then for any $ A\subseteq P$, let $ A'$ be the set of finite joins of elements of $ A$, then $ A'$ is a directed set, and $ \langle A\rangle=\downarrow\!\!A'$.

Dually, an order filter (or simply a filter) in $ P$ is a non-empty subset $ F$ which is both an upper set and a filtered set (every pair of elements in $ F$ has a lower bound in $ F$). A principal filter is a filter of the form $ \uparrow\!\!x$ for some $ x\in P$.

Remark. This is a generalization of the notion of a filter in a set. In fact, both ideals and filters are generalizations of ideals and filters in semilattices and lattices.

Examples in a Semilattice

A subset $ I$ in an upper semilattice $ P$ is a semilattice ideal if

  1. if $ a,b\in I$, then $ a\vee b\in I$ (condition for being an upper subsemilattice)
  2. if $ a\in I$ and $ b\le a$, then $ b\in I$

Then the two definitions are equivalent: if $ P$ is an upper semilattice, then $ I\subseteq P$ is a semilattice ideal iff $ I$ is an order ideal of $ P$: if $ I$ is a semilattice ideal, then $ I$ is clearly a lower and directed (since $ a\vee b$ is an upper bound of $ a$ and $ b$); if $ I$ is an order ideal, then condition 2 of a semilattice ideal is satisfied. If $ a,b\in I$, then there is a $ c\in I$ that is an upper bound of $ a$ and $ b$. Since $ I$ is lower, and $ a\vee b\le c$, we have $ a\vee b\in I$.

Going one step further, we see that if $ P$ is a lattice, then a lattice ideal is exactly an order ideal: if $ I$ is a lattice ideal, then it is clearly an upper subsemilattice, and if $ b\le a\in I$, then $ b=a\wedge b\in I$ also, so that $ I$ is a semilattice ideal. On the other hand, if $ I$ is a semilattice ideal, then $ I$ is an upper subsemilattice, as well as a lower subsemilattice, for if $ a\in I$, then $ a\wedge b\in I$ as well since $ a\wedge b\le a$. This shows that $ I$ is a lattice ideal.

Dually, we can define a filter in a lower semilattice, which is equivalent to an order filter of the underly poset. Going one step futher, we also see that a lattice filter in a lattice is an order filter of the underlying poset.

Remark. An alternative but equivalent characterization of a semilattice ideal $ I$ in an upper semilattice $ P$ is the following: $ a,b\in I$ iff $ a\vee b\in I$.



"order ideal" is owned by CWoo.
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See Also: filter, lattice filter, lattice ideal

Other names:  filter, ideal
Also defines:  order filter, semilattice ideal, semilattice filter, subsemilattice, principal ideal, principal filter

Attachments:
ideal completion of a poset (Definition) by CWoo
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Cross-references: characterization, lattice filter, lower semilattice, lattice ideal, lattice, equivalent, definitions, lower bound, filtered set, upper set, joins, finite, upper semilattice, iff, ideal generated by, upper bound, directed set, lower set, subset, poset
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This is version 8 of order ideal, born on 2007-04-30, modified 2007-07-25.
Object id is 9305, canonical name is OrderIdeal.
Accessed 2320 times total.

Classification:
AMS MSC: 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)
 06A12 (Order, lattices, ordered algebraic structures :: Ordered sets :: Semilattices)
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