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. 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. 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. 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.