# Maximal and minimal element in preordered set

Generally the notion of maximal and minimal element is defined in a partially ordered set (binary relation is reflexive, antisymmetric and transitive).

A preorder is a binary relation that is reflexive and transitive.

Can we define the notion of maximal and minimal element in a preordered set?

• Sure you can, the same way you do in a poset: $x$ is maximal if $x\leq y$ implies $x=y$. However, it will not generally have the properties you might want. For example, in the total pre-order on a set with more than one element, there are no maximal elements. Apr 5, 2019 at 5:54
• @ArturoMagidin Why not define $x$ is maximal if $x\le y$ implies $y\le x$? I.e., $x$ is maximal if there is no $y$ such that $x\lt y$.
– bof
Apr 5, 2019 at 8:49
• @bof: again, you can, but it may not have the properties that you "expect"/"hope". The real question is: what do you want these elements to do in a preorder (and more especially, do you want them to behave in certain ways when you mod out by the equivalence to get an order)? Apr 5, 2019 at 13:55
• @AlbertoTakase The standard definition of $x\lt y$ for a quasi-ordering $\le$ (e.g., the ordering of linear order types by embeddability) is $x\le y\land y\not\le x$.
– bof
Apr 9, 2019 at 15:40
• I appreciate the correction. I now understand ${\prec}:={\preceq}\setminus{\approx}$, where ${\approx}=\{(x,y):x\preceq y\preceq x\}$. I guess the notion of "minimal" for preorders $\preceq$ and binary relations $R$ (in the context of well-foundedness) are different. Apr 9, 2019 at 16:07

Definition. Let $$X$$ be a set. Let $$\preceq$$ be a preorder in $$X$$. Fix $$S\subseteq X$$.
A $$\preceq$$-minimal-element of $$S$$ is an element $$t$$ of $$S$$ such that $$(\forall s\in S)[s\preceq t\:\Rightarrow\:s=t]$$.
A $$\preceq$$-maximal-element of $$S$$ is an element $$t$$ of $$S$$ such that $$(\forall s\in S)[t\preceq s\:\Rightarrow\:t=s]$$.