# User:Kompik/sandbox

## Orderings

Sometimes we will write, for a relation ${\displaystyle R}$, ${\displaystyle xRy}$ instead of ${\displaystyle (x,y)\in R}$. In this chapter we will deal with ordering -- relations with special properties and we will denote these relations usally ${\displaystyle \leq }$. In fact, the definition of ordering reminds properties of the usual relation ${\displaystyle \leq }$ on numbers.

A relation R on set A is called

• reflexive iff aRa for any ${\displaystyle a\in A}$;
• antisymmetric iff if aRb and bRa implies a = b for any ${\displaystyle a,b\in A}$;
• transitive iff aRb and bRc implies aRc for any ${\displaystyle a,b\in A}$.

partial order

Note about weak < and strict partial order.

totally ordered set linearly ordered set (total order, linear order), chain

antichain

Examples: ${\displaystyle \subseteq }$, |

minimal element

smallest element

well-ordering

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