Semiorder

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 $$u(x)\le u(y)-1$$. 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:
 * Whenever two disjoint pairs of elements are comparable, for instance as $$w<x$$ and $$y<z$$, there must be an additional comparison among these elements, because $$u(w)\le u(y)$$ would imply $$w<z$$ while $$u(w)\ge u(y)$$ would imply $$y<x$$. Therefore, it is impossible to have two mutually incomparable two-point linear orders.
 * If three elements form a linear ordering $$w<x<y$$, then every fourth point $$z$$ must be comparable to at least one of them, because $$u(z)\le u(x)$$ would imply $$z<y$$ while $$u(z)\ge u(x)$$ would imply $$w<z$$, in either case showing that $$z$$ is comparable to $$w$$ or to $$y$$. So it is impossible to have a three-point linear order with a fourth incomparable point.

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.

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 $$[u(x),u(x)+1]$$, 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 $$(\ell _{i},\ell _{i}+1)$$.

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