Q-orders with no unit on Q

For a usual commutative quantale Q (does not necessarily have a unit), we propose a definition of Q-ordered sets by introducing a kind of self-adaptive self-reflexivity. We study their completeness and the related Q-modules of complete lattices. The main result is that, the complete Q-ordered sets and the Q-modules of complete lattices are categorical isomorphic.

Optimal Pricing Under Multiple-Discrete Customer Choices and Diminishing Return of Consumption

Customers make purchase decisions based on the attributes of the products offered and their prices. While the customer selects only one unit of a product in some settings, she purchases multiple products and even possibly multiple units of each product in other settings. Although several studies in the literature have addressed the former case, little attention has been paid to the latter case, this paper’s subject. In this paper, the authors consider the customer's problem and show that the set of products she purchases is one of the ordered sets based on the product attribute and prices. This paper shows that the firm's optimal pricing problem can be solved efficiently based on another ordering among the products.

The structure and organization of headache differential diagnoses: A Pilot Study of Subset Relationships between Differentials in ICHD3

Objective: Differential diagnosis is fundamental to medicine. Using DiffNet, a differential diagnosis generator, as a model we studied the structure and organization of how collections of diagnose (i.e. sets of diagnoses) are related in the ICHD3. Specifically, we asked: Which sets of differential diagnoses are subsets of each other? What is the minimum number of sets of differential diagnoses that encompass all ICHD3 codes? Furthermore, we explored the clinical and theoretical implication of these answers. Methods: DiffNet is a freely distributed differential diagnosis generator for headaches using graph theoretical properties of ICHD3. For each ICHD3 diagnosis, we generated a set of differential diagnoses using DiffNet. We then determined algorithmically the set/subset relationship between these sets. We also determined the smallest list of ICHD3 diagnosis whose differential diagnoses would encompass the totality of ICHD3 diagnoses. Results: All ICHD3 diagnoses can be represented by a minimum of 92 differential diagnosis sets. Differential diagnosis sets for 10 of the 14 first digit subcategories of ICHD3 are represented by more than one differential diagnosis sets. Fifty-one of the 93 differential diagnosis sets contain multiple subset relationships; the remaining 42 do not enter into any set/subset relationship with other differential diagnosis sets. Finally, we included a hierarchical presentation of differential diagnosis sets in ICHD3 according to DiffNet. Conclusion: We propose a way of interpreting headache differential diagnoses as partial ordered sets (i.e. poset). For clinicians, fluency with the 93 diagnoses and their differential put forth here implies a complete description of ICHD3. On a theoretical level, interpreting ICHD3 differential diagnosis as poset, allows researchers to translate differential diagnoses sets topologically, algebraically, and categorically.

A birational lifting of the Stanley-Thomas word on products of two chains

The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key property enjoyed by the rowmotion operator on certain finite partially-ordered sets is homomesy, where the average value of a statistic is the same for all orbits. To prove refined versions of homomesy in the product of two chain posets, J. Propp and the second author used an equivariant bijection discovered (less formally) by R. Stanley and H. Thomas. We explore the lifting of this "Stanley--Thomas word" to the piecewise-linear, birational, and noncommutative realms. Although the map is no longer a bijection, so cannot be used to prove periodicity directly, it still gives enough information to prove the homomesy at the piecewise-linear and birational levels (a result previously shown by D. Grinberg, S. Hopkins, and S. Okada). Even at the noncommutative level, the Stanley--Thomas word of a poset labeling rotates cyclically with the lifting of antichain rowmotion. Along the way we give some formulas for noncommutative antichain rowmotion that we hope will be first steps towards proving the conjectured periodicity at this level. Comment: 20 pages, 6 figures

Smooth approximations by continuous choice-functions

We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake we prove that the theory of densely ordered sets with generic predicates is ℵ0- categorical. A model of the theory and a particular continuous choice-function are constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a byproduct, other functions with surprising properties can be constructed.

Order continuity from a topological perspective

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups, and partially ordered vector spaces, respectively. An order topology is introduced such that in the latter two settings under mild conditions order continuity is a topological property. We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.