✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✓ ✗ ✗ ✗ ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✓ ✗ ✗ ✗ ✓ ✓ ✓ ✗ ✗ ✗ ✓ ✗ ✗ ✓ ✓ ✗ ✓ ✗ ✗ ✓ ✗ ✗ ✓ ✗ ✗ ✗ ✓ A ' ✓' indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require. In, especially in, a preorder or quasiorder is a that is.

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Preorders are more general than and (non-strict), both of which are special cases of a preorder. An preorder is a partial order, and a preorder is an. The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily anti-symmetric nor. Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.

In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or ≲ is used instead of ≤. To every preorder, there corresponds a, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain.

A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components. • for n = 3: • 1 partition of 3, giving 1 preorder • 3 partitions of 2 + 1, giving 3 × 3 = 9 preorders • 1 partition of 1 + 1 + 1, giving 19 preorders I.e., together, 29 preorders. • for n = 4: • 1 partition of 4, giving 1 preorder • 7 partitions with two classes (4 of 3 + 1 and 3 of 2 + 2), giving 7 × 3 = 21 preorders • 6 partitions of 2 + 1 + 1, giving 6 × 19 = 114 preorders • 1 partition of 1 + 1 + 1 + 1, giving 219 preorders I.e., together, 355 preorders. Interval [ ] For a ≲ b, the [ a, b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b.

It contains at least the points a and b. One may choose to extend the definition to all pairs ( a, b). The extra intervals are all empty. Using the corresponding strict relation '.

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