The Hasse diagram of a semiorder. Two items are comparable when their vertical coordinates differ by at least one unit (the spacing between solid blue lines).

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.

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 ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ represent three quantities of the same material, and that ${\displaystyle x}$ is larger than ${\displaystyle z}$ by the smallest amount that is perceptible as a difference, while ${\displaystyle y}$ is halfway between the two of them. Then, a person who desires more of the material would prefer ${\displaystyle x}$ to ${\displaystyle z}$, but would not have a preference between the other two pairs. In this example, ${\displaystyle x}$ and ${\displaystyle y}$ are incomparable in the preference ordering, as are ${\displaystyle y}$ and ${\displaystyle z}$, but ${\displaystyle x}$ and ${\displaystyle z}$ 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 ${\displaystyle u}$ be any utility function that maps the objects to be compared (a set ${\displaystyle X}$) 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 ${\displaystyle <}$ on the objects, by setting ${\displaystyle x<y}$ whenever ${\displaystyle u(x)\leq u(y)-1}$. Then ${\displaystyle (X,<)}$ 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]

Axiomatics

Forbidden: two mutually incomparable two-point linear orders

Forbidden: three linearly ordered points and a fourth incomparable point

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 ${\displaystyle w<x}$ and ${\displaystyle y<z}$, there must be an additional comparison among these elements, because ${\displaystyle u(w)\leq u(y)}$ would imply ${\displaystyle w<z}$ while ${\displaystyle u(w)\geq u(y)}$ would imply ${\displaystyle y<x}$. Therefore, it is impossible to have two mutually incomparable two-point linear orders.^[3]
• If three elements form a linear ordering ${\displaystyle w<x<y}$, then every fourth point ${\displaystyle z}$ must be comparable to at least one of them, because ${\displaystyle u(z)\leq u(x)}$ would imply ${\displaystyle z<y}$ while ${\displaystyle u(z)\geq u(x)}$ would imply ${\displaystyle w<z}$, in either case showing that ${\displaystyle z}$ is comparable to ${\displaystyle w}$ or to ${\displaystyle y}$. 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 ${\displaystyle n}$ 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 ${\displaystyle O(n^{2})}$, where the ${\displaystyle O}$ 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 ${\displaystyle (X,<)}$ (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. Fishburn (1973) supplies a precise characterization of the semiorders that may be defined numerically.^[7]

Relation to other kinds of order

Combinatorial enumeration

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

$${\displaystyle {\frac {1}{n+1}}{\binom {2n}{n}},}$$

while the number of semiorders on ${\displaystyle n}$ labeled items is given by the sequence^[14]

Other results

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 ${\displaystyle x}$ and ${\displaystyle y}$ such that ${\displaystyle x}$ appears earlier than ${\displaystyle y}$ in between 1/3 and 2/3 of the linear extensions of the semiorder.^[3]

The set of semiorders on an ${\displaystyle n}$-element set is well-graded: if two semiorders on the same set differ from each other by the addition or removal of ${\displaystyle k}$ order relations, then it is possible to find a path of ${\displaystyle k}$ 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.^[17]

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

Notes

References

Further reading

• Pirlot, M.; Vincke, Ph. (1997), Semiorders: Properties, representations, applications, Theory and Decision Library. Series B: Mathematical and Statistical Methods, vol. 36, Dordrecht: Kluwer Academic Publishers Group, ISBN 0-7923-4617-3, MR 1472236.