# connected poset

Let $P$ be a poset. Write $a\u27c2b$ if either $a\le b$ or $b\le a$. In other words, $a\u27c2b$ if $a$ and $b$ are comparable^{}. A poset $P$ is said to be *connected ^{}* if for every pair $a,b\in P$, there is a finite sequence

^{}$a={c}_{1},{c}_{2},\mathrm{\dots},{c}_{n}=b$, with each ${c}_{i}\in P$, such that ${c}_{i}\u27c2{c}_{i+1}$ for each $i=1,2,\mathrm{\dots},n-1$.

For example, a poset with the property that any two elements are either bounded from above or bounded from below is a connected poset. In particular, every semilattice is connected. A fence is always connected. If $P$ has more than one element and contains an element that is both maximal and minimal^{}, then it is not connected. A *connected component ^{}* in a poset $P$ is a maximal connected subposet. In the last example, the maximal-minimal point is a component in $P$. Any poset can be written as a disjoint union

^{}of its components.

It is true that a poset is connected if its corresponding Hasse diagram^{} is a connected graph. However, the converse^{} is not true. Before we see an example of this, let us recall how to construct a Hasse diagram from a poset $P$. The diagram so constructed is going to be an undirected graph (since this is all we need in our discussion). Draw an edge between $a,b\in P$ if either $a$ covers $b$ or $b$ covers $a$. Let us denote this relation^{} between $a$ and $b$ by $a\asymp b$. Let $E$ be the collection^{} of all these edges. Then $G=(P,E)$ is a graph where elements of $P$ serve as vertices and $E$ as the constructed edges. From this construction, one sees that a finite path exists between $a,b\in V(G)=P$ if there is a finite sequence $a={d}_{0},{d}_{1},\mathrm{\dots},{d}_{m}=b$, with each ${d}_{i}\in V(G)$, such that ${d}_{i}\asymp {d}_{i+1}$ for $i=1,\mathrm{\dots},m-1$. In other words, $a$ and $b$ can be “joined” by a finite number of edges, such that $a$ is a vertex on the first edge and $b$ is the vertex on the last edge.

As promised, here is an example of a connected poset whose underlying Hasse diagram is not connected. take the real line $\mathbb{R}$ with $\mathrm{\infty}$ adjoined to the right (meaning every element $r\in \mathbb{R}$ is less than or equal to $\mathrm{\infty}$). Then the resulting poset is connected, but its underlying Hasse diagram is not, since no element in $\mathbb{R}$ can be joined to $\mathrm{\infty}$ by a finite path.

Title | connected poset |
---|---|

Canonical name | ConnectedPoset |

Date of creation | 2013-03-22 17:08:31 |

Last modified on | 2013-03-22 17:08:31 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 06A07 |

Related topic | ConnectedGraph |

Defines | connected |

Defines | connected component |