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

A relation R on set A is called
 * reflexive iff aRa for any $$a\in A$$;
 * antisymmetric iff if aRb and bRa implies a = b for any $$a,b\in A$$;
 * transitive iff aRb and bRc implies aRc for any $$a,b\in A$$.

partial order

Note about weak < and strict partial order.

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

antichain

Examples: $$\subseteq$$, |

minimal element

smallest element

well-ordering

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