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 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.

Utility theory

The original motivation for introducing semiorders was to model human preferences without assuming that incomparability is a transitive relation. For instance, suppose that x, y, and z represent three quantities of the same material, and that x is larger than z by the smallest amount that is perceptible as a difference, while y is halfway between the two of them. Then, a person who desires more of the material would prefer x to z, but would not have a preference between the other two pairs. In this example, x and y are incomparable in the preference ordering, as are y and z, but x and z are comparable, so incomparability does not obey the transitive law. To model this mathematically, suppose that objects are given numerical utility values, by letting u be any utility function that maps the objects to be compared (a set X) 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 x<y whenever. Then (X,<) forms a semiorder. 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.

Axiomatics

A semiorder, defined from a utility function as above, is a partially ordered set with the following two properties: 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. 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. If a semiorder on n 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 O(n^2), where the O is an instance of big O notation. 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 (X,<) (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. supplies a precise characterization of the semiorders that may be defined numerically.

Relation to other kinds of order

Partial orders

One may define a partial order (X,\le) from a semiorder (X,<) by declaring that x\le y whenever either x<y or x=y. Of the axioms that a partial order is required to obey, reflexivity (x\le x) follows automatically from this definition. Antisymmetry (if x\le y and y\le x then x=y) follows from the first semiorder axiom. Transitivity (if x\le y and y\le z then x\le z) follows from the second semiorder axiom. Therefore, the binary relation (X,\le) defined in this way meets the three requirements of a partial order that it be reflexive, antisymmetric, and transitive. Conversely, suppose that (X,\le) is a partial order that has been constructed in this way from a semiorder. Then the semiorder may be recovered by declaring that x<y whenever x\le y and x\ne y. 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 (X,<) 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 ).

Weak orders

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.

Interval orders

The semiorder defined from a utility function u may equivalently be defined as the interval order defined by the intervals, 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.

Quasitransitive relations

According to Amartya K. Sen, semi-orders were examined by Dean T. Jamison and Lawrence J. Lau and found to be a special case of quasitransitive relations. In fact, every semiorder is quasitransitive, and quasitransitivity is invariant to adding all pairs of incomparable items. 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.

Combinatorial enumeration

The number of distinct semiorders on n unlabeled items is given by the Catalan numbers while the number of semiorders on n labeled items is given by the sequence

Other results

Any finite semiorder has order dimension at most three. 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. 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 x and y such that x appears earlier than y in between 1/3 and 2/3 of the linear extensions of the semiorder. The set of semiorders on an n-element set is well-graded: if two semiorders on the same set differ from each other by the addition or removal of k order relations, then it is possible to find a path of k 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. The incomparability graphs of semiorders are called indifference graphs, and are a special case of the interval graphs.