In order theory, a branch of mathematics, a **semiorder** is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given margin of error are deemed incomparable. Semiorders were introduced and applied in mathematical psychology by Duncan Luce (1956) as a model of human preference. They generalize strict weak orderings, in which items with equal scores may be tied but there is no margin of error. They are a special case of partial orders and of interval orders, and can be characterized among the partial orders by additional axioms, or by two forbidden four-item suborders.

The original motivation for introducing semiorders was to model human preferences without assuming that incomparability is a transitive relation. For instance, suppose that , , and represent three quantities of the same material, and that is larger than by the smallest amount that is perceptible as a difference, while is halfway between the two of them. Then, a person who desires more of the material would prefer to , but would not have a preference between the other two pairs. In this example, and are incomparable in the preference ordering, as are and , but and are comparable, so incomparability does not obey the transitive law.^{[1]}

To model this mathematically, suppose that objects are given numerical utility values, by letting be any utility function that maps the objects to be compared (a set ) to real numbers. Set a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each other are declared incomparable, and define a binary relation on the objects, by setting whenever . Then forms a semiorder.^{[2]}
If, instead, objects are declared comparable whenever their utilities differ, the result would be a strict weak ordering, for which incomparability of objects (based on equality of numbers) would be transitive.^{[1]}

A semiorder, defined from a utility function as above, is a partially ordered set with the following two properties:^{[3]}

- Whenever two disjoint pairs of elements are comparable, for instance as and , there must be an additional comparison among these elements, because would imply while would imply . Therefore, it is impossible to have two mutually incomparable two-point linear orders.
^{[3]} - If three elements form a linear ordering , then every fourth point must be comparable to at least one of them, because would imply while would imply , in either case showing that is comparable to or to . So it is impossible to have a three-point linear order with a fourth incomparable point.
^{[3]}

Conversely, every finite partial order that avoids the two forbidden four-point orderings described above can be given utility values making it into a semiorder.
^{[4]} Therefore, rather than being a consequence of a definition in terms of utility, these forbidden orderings, or equivalent systems of axioms, can be taken as a combinatorial definition of semiorders.^{[5]} If a semiorder on elements is given only in terms of the order relation between its pairs of elements, obeying these axioms, then it is possible to construct a utility function that represents the order in time , where the is an instance of big O notation.^{[6]}

For orderings on infinite sets of elements, the orderings that can be defined by utility functions and the orderings that can be defined by forbidden four-point orders differ from each other. For instance, if a semiorder (as defined by forbidden orders) includes an uncountable totally ordered subset then there do not exist sufficiently many sufficiently well-spaced real-numbers for it to be representable by a utility function. Fishburn (1973) supplies a precise characterization of the semiorders that may be defined numerically.^{[7]}

One may define a partial order from a semiorder by declaring that whenever either or . Of the axioms that a partial order is required to obey, reflexivity () follows automatically from this definition. Antisymmetry (if and then ) follows from the first semiorder axiom. Transitivity (if and then ) follows from the second semiorder axiom. Therefore, the binary relation defined in this way meets the three requirements of a partial order that it be reflexive, antisymmetric, and transitive.

Conversely, suppose that is a partial order that has been constructed in this way from a semiorder. Then the semiorder may be recovered by declaring that whenever and . Not every partial order leads to a semiorder in this way, however: The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, but the others do not. A partial order that includes four elements forming two two-element chains would lead to a relation that violates the second semiorder axiom, and a partial order that includes four elements forming a three-element chain and an unrelated item would violate the third semiorder axiom (cf. pictures in section #Axiomatics).

Every strict weak ordering < is also a semi-order. More particularly, transitivity of < and transitivity of incomparability with respect to < together imply the above axiom 2, while transitivity of incomparability alone implies axiom 3. The semiorder shown in the top image is not a strict weak ordering, since the rightmost vertex is incomparable to its two closest left neighbors, but they are comparable.

The semiorder defined from a utility function may equivalently be defined as the interval order defined by the intervals ,^{[8]} so every semiorder is an example of an interval order.
A relation is a semiorder if, and only if, it can be obtained as an interval order of unit length intervals .

According to Amartya K. Sen,^{[9]} semi-orders were examined by Dean T. Jamison and Lawrence J. Lau^{[10]} and found to be a special case of quasitransitive relations. In fact, every semiorder is quasitransitive,^{[11]} and quasitransitivity is invariant to adding all pairs of incomparable items.^{[12]} Removing all non-vertical red lines from the topmost image results in a Hasse diagram for a relation that is still quasitransitive, but violates both axiom 2 and 3; this relation might no longer be useful as a preference ordering.

The number of distinct semiorders on unlabeled items is given by the Catalan numbers^{[13]}

while the number of semiorders on labeled items is given by the sequence

Any finite semiorder has order dimension at most three.^{[15]}

Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of linear extensions are semiorders.^{[16]}

Semiorders are known to obey the 1/3–2/3 conjecture: in any finite semiorder that is not a total order, there exists a pair of elements and such that appears earlier than in between 1/3 and 2/3 of the linear extensions of the semiorder.^{[3]}

The set of semiorders on an -element set is *well-graded*: if two semiorders on the same set differ from each other by the addition or removal of order relations, then it is possible to find a path of steps from the first semiorder to the second one, in such a way that each step of the path adds or removes a single order relation and each intermediate state in the path is itself a semiorder.^{[17]}

The incomparability graphs of semiorders are called indifference graphs, and are a special case of the interval graphs.^{[18]}