# Is this a partially ordered set?

I was wondering if partially ordered sets could have loops in their diagrams. For example isn't the $S=\{1,2,3\}$ and relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(3,1)\}$ a partially ordered set that has a cycle? $R$ is reflexive, antisymmetric and transitive.

Transitivy fails for your relation $R$.
• @Celeritas Exactly. It also fails because $(2,3), (3,1)\in R$, but $(2,1)\not \in R$. Nov 24, 2013 at 11:55
• @Celeritas In a partial order, $a<b\iff a\leq b \land a\neq b$, replacing $a$ and $b$ with $x$ yields $x<x\iff x\leq x \land \color{red}{x\neq x}$. Is this clear? Nov 24, 2013 at 12:55
• @Celeritas $\prec$ is used for covering. Notice the name of the command \prec, it comes from preciding which is the same as covering. In my comment above $<$ intuitively is 'less than', yes and it is different from $\prec$. What exactly aren't you understanding at the moment? Nov 24, 2013 at 13:36
• @Celeritas It's what denoted here as '$<\cdot$', $a\prec b$ means '$b$ covers $a$'. Nov 24, 2013 at 14:37