Susan H. Gensemer

Continuous semiorder representations

Given 〈X,≻〉, where X is a topologically connected space and ≻ is an asymmetric binary relation, necessary and sufficient conditions are presented for the existence of a continuous representation of the form, 〈u, δ〉; that is, for x,y∈X, x≻y if and only if u(x)>u(y)+δ where u:X→R is continuous and δ is a strictly positive real number. The results are related to existing results for numerical representations of interval orders and semiorders.

On numerical representations of semiorders

Abstract Conditions on a semiorder, 〈 X ,≻〉 (where X is a subset of Euclidean space), are presented under which the semiorder can be represented by a continuous, real-valued function, u , along with a strictly positive scalar, δ . That is, for x,y ϵ X, x ≻ y if and only if u ( x )> u ( y )+ δ . The results are related to existing results for numerical representations of interval orders.

On relationships between numerical representations of interval orders and semiorders

Abstract The extent to which continuous numerical representations of interval orders are unique is considered. Apair of continuous, real-valued functions, , represents an interval order, >, provided that for x, y ϵ X, x > y if and only if u(x) > v(y). Relationships which necessarily hold between any two such numerical representations are presented and a method by which one continuous representation can be derived from another is described. Similar considerations are made for special forms of continuous numerical representations of semiorders.

Partial Groupoid Embeddings in Semigroups

We examine a number of axiom systems guaranteeing the embedding of a partial groupoid into a semigroup. These include the Tamari symmetric partial groupoid and the Gensemer/Weinert equidivisible partial groupoid, provided they satisfy an additional axiom, weak associativity. Both structures share the one mountain property. More embedding results for partial groupoids into other types of algebraic structures are presented as well.

Intransitive indifference and incomparability

Abstract This paper presents the concept of an incomplete semiorder and the result that such a structure can be extended to a semiorder. Corresponding results exist for interval orders. An economic application is provided.

Revealed preference and intransitive indifference

Abstract This paper approaches revealed preference from an assumption of a type of bounded rationality on the part of the consumer. We assume that the individual may have discrimination problems with regard to alternative choices (specifically, we assume that the underlying preferences of the consumer are described by a semiorder). The main result of this paper is the description of a revealed preference condition which is satisfied by a set of partial choice data if and only if the partial choice function can be rationalized by semiorder preferences.

An efficient algorithm for voting sequences

Abstract The chaos theorems show that given almost any alternatives x and y, there exists voting sequence from x to y. However, proofs of such results have been purely existential; that is, there is no algorithm by which such a voting path can be constructed. In this paper, we present such an algorithm for one standard example. Furthermore, it is shown that the algorithm has the property that the voting sequence involves the fewest possible number of steps.