Hasse diagram of finite linearly ordered set

Question

What form does the Hasse diagram of a finite linearly ordered set take? I think the linearly order set is nothing but totally ordered set which usually takes lattice form since every element is comparable. However am confused if it is the same or something different. My textbook says it takes linear path form.

Answer 1

Your textbook is correct. Linearly ordered set and totally ordered set are the same thing. If a linearly ordered set has $n$ elements, its Hasse diagram is a vertical path with $n$ nodes and $n-1$ edges. For $n=5$, for instance, it looks like this:

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The set $\{0,1,2,3,4\}$ with the usual linear order $\le$ is an example of a linearly ordered set for which this is the Hasse diagram. Another example is the set

$$\big\{\varnothing,\{0\},\{0,1\},\{0,1,2\},\{0,1,2,3\}\big\}$$

with the linear order $\subseteq$.