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Orderings[edit]

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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See also[edit]

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