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| {{{#!latex2 A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair $P=\left<X,\sqsubseteq \right>$, where $X$ is called the ground set of $P$ and $\sqsubseteq$ is the partial order of $P$. |
A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair $$P=\left<X,\sqsubseteq \right>$$, where $$X$$ is called the ground set of $$P$$ and $$\sqsubseteq$$ is the partial order of $$P$$. |
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| An element $u$ in a partially ordered set $\left< X,\sqsubseteq \right>$ is said to be an upper bound for a subset $S$ of $X$ if for every $s \in S$, we have $s \sqsubseteq u$. Similarly, a lower bound for a subset $S$ is an element $l$ such that for every $s \in S$, $l \sqsubseteq s$. If there is an upper bound and a lower bound for X, then the poset $\left< X,\sqsubseteq \right>$ is said to be bounded. }}} |
An element $$u$$ in a partially ordered set $$\left< X,\sqsubseteq \right>$$ is said to be an upper bound for a subset $$S$$ of $$X$$ if for every $$s \in S$$, we have $$s \sqsubseteq u$$. Similarly, a lower bound for a subset $$S$$ is an element $$l$$ such that for every $$s \in S$$, $$l \sqsubseteq s$$. If there is an upper bound and a lower bound for X, then the poset $$\left< X,\sqsubseteq \right>$$ is said to be bounded. |
Partially Ordered Set
The following was adapted from Wolfram's site:
A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair $$P=\left<X,\sqsubseteq \right>$$, where $$X$$ is called the ground set of $$P$$ and $$\sqsubseteq$$ is the partial order of $$P$$.
An element $$u$$ in a partially ordered set $$\left< X,\sqsubseteq \right>$$ is said to be an upper bound for a subset $$S$$ of $$X$$ if for every $$s \in S$$, we have $$s \sqsubseteq u$$. Similarly, a lower bound for a subset $$S$$ is an element $$l$$ such that for every $$s \in S$$, $$l \sqsubseteq s$$. If there is an upper bound and a lower bound for X, then the poset $$\left< X,\sqsubseteq \right>$$ is said to be bounded.
See PartialOrder