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A strict total order on a set is a strict partial order on in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :
For the sorting to be unique, these two are restricted to a total order and a strict total order, respectively. Sorting n-tuples (depending on context also called e.g. records consisting of fields) can be done based on one or more of its components. More generally objects can be sorted based on a property.
A strict weak order that is trichotomous is called a strict total order. [14] The total preorder which is the inverse of its complement is in this case a total order . For a strict weak order < {\displaystyle \,<\,} another associated reflexive relation is its reflexive closure , a (non-strict) partial order ≤ . {\displaystyle \,\leq .}
Total orders, orderings that specify, for every two distinct elements, which one is less than the other; Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities) Well-orders, total orders in which every non-empty subset has a least element
Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty ...
Partial order – an antisymmetric preorder; Total preorder – a connected (formerly called total) preorder; Equivalence relation – a symmetric preorder; Strict weak ordering – a strict partial order in which incomparability is an equivalence relation; Total ordering – a connected (total), antisymmetric, and transitive relation
In order to prove termination, a reduction ordering (>) must be found with respect to which the term x*(y+z) is greater than the term (x*y)+(x*z). This is not trivial, since the former term contains both fewer function symbols and fewer variables than the latter.
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