What is the definition of first/last element in a poset?

Question

I have read the terms first element/last elements in the context of a basic course in set theory.

When I learned about posets I didn't encounter those terms. I tried looking up the definitions but I didn't find them.

Can someone please write down the definitions for first/last element in a poset ?

Answer 1

The first element of a poset $\langle P,\le\rangle$ is simply the unique minimum element of $P$, if there is one: $p_0$ is the first element of $P$ if $p_0\le p$ for all $p\in P$. Similarly, the last element of $P$ is the unique maximum element of $P$, if there is one: $p_1$ is the last element of $P$ if $p\le p_1$ for all $p\in P$.

A poset need not have a first or last element.

Answer 2

If by "first" element you mean "an element preceding all others", then there need not be one (similarly for "last" element).

Consider the sets $\{1\}$, $\{2\}$, $\{3\}$, $\{1,2\}$, and $\{1,3\}$ ordered by inclusion. There is no first or last element, but there are three minimal elements (an element with no predecessor) and two maximal elements (an element with no successor).

In the case where there is an element preceding all others, it is generally called a minimum element. Similarly, the element that is preceded by all others is called a maximum element.

Answer 3

A poset need not have a first or last element.

• An element $a_1$ is first (the unique minimum) in a poset $P$ if $a_1 \le a\;\;\forall a \in P$.
• An element $a_n$ is last (the unique maximum) in a poset P if $a \le a_n \;\;\forall a \in P$.

(1) There may not be a "minimum" (first) nor "maximum" (last) element in a poset.

(2) You might have a unique minimum (first) element, but no maximum element (last) in a poset.

(3) Likewise, a poset may not have a minimum "first" element, but have a maximum ("last") element.

(4) And if there is a unique minimum and a unique maximum element in the poset, then those are the first and last elements in the poset, respectively.

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Exercise: try to find an "example" poset that models/represents each of these four possibilities) (one per possibility)!