ordering that resembles weak orderings, but is better modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another, or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they are within the margin of error of each other. However, if candidate is statistically tied with and is statistically tied with it might still be possible for to be clearly better than so being tied is not in this case a transitive relation. Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings. Axiomatizations Suppose throughout that is a homogeneous binary relation on a set (that is, is a subset of ) and as usual, write and say that or if and only if Strict weak orderings Preliminaries on incomparability and transitivity of incomparability Two elements and of are said to be with respect to if neither is true. Incomparability with respect to is itself a homogeneous symmetric relation on that is reflexive if and only if is irreflexive (meaning that is always false), which may be assumed so that transitivity is the only property this "incomparability relation" needs in order to be an equivalence relation. Define also an induced homogeneous relation on by declaring that where importantly, this definition is necessarily the same as: if and only if Two elements are incomparable with respect to if and only if are with respect to (or less verbosely, ), which by definition means that both are true. The relation "are incomparable with respect to " is thus identical to (that is, equal to) the relation "are -equivalent" (so in particular, the former is transitive if and only if the latter is). When is irreflexive then the property known as "transitivity of incomparability" (defined below) is the condition necessary and sufficient to guarantee that the relation "are -equivalent" does indeed form an equivalence relation on When this is the case, it allows any two elements satisfying to be identified as a single object (specifically, they are identified together in their common equivalence class). Definition A strict weak ordering on a set is a strict partial order on for which the induced on by is a transitive relation. Explicitly, a strict weak order on is a homogeneous relation on that has all four of the following properties: : For