{"history":[],"article":{"id":"359fa2e3-b72c-4de9-adff-1a1f0aa5e729","title":"Semiorder","slug":"semiorder","content":":::summary\n:::\n\n# \nIn [order theory](https://bliptext.com/articles/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](https://bliptext.com/articles/margin-of-error) are deemed [incomparable](https://bliptext.com/articles/comparability).\nSemiorders were introduced and applied in [mathematical psychology](https://bliptext.com/articles/mathematical-psychology) by as a model of human preference.\nThey generalize [strict weak orderings](https://bliptext.com/articles/strict-weak-ordering), in which items with equal scores may be tied but there is no margin of error.\nThey are a special case of [partial orders](https://bliptext.com/articles/partial-order) and of [interval orders](https://bliptext.com/articles/interval-order), and can be characterized among the [partial orders](https://bliptext.com/articles/partial-order) by additional [axioms](https://bliptext.com/articles/axiom), or by two forbidden four-item suborders.\n# [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). | upload.wikimedia.org/wikipedia/commons/3/33/Semiorder.svg]\n\n# Utility theory\nThe original motivation for introducing semiorders was to model human preferences without assuming that incomparability is a [transitive relation](https://bliptext.com/articles/transitive-relation).\nFor 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.\nThen, a person who desires more of the material would prefer x to z, but would not have a preference between the other two pairs.\nIn 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.\nTo model this mathematically, suppose that objects are given numerical [utility values](https://bliptext.com/articles/utility-function), by letting u be any [utility function](https://bliptext.com/articles/utility-function) that maps the objects to be compared (a set X) to [real numbers](https://bliptext.com/articles/real-number).\nSet 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

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