Pre-ordered Set Research Articles

Educational Value of an IBD Inpatient Order Set: 2016 ACG Presidential Poster Award

Introduction: Inflammatory bowel disease (IBD) is an autoimmune disorder of the GI tract which often requires hospitalizations and at times surgical interventions. We implemented a computerized provider order entry (CPOE) set based on quality indicators approved by the Crohn's and Colitis Foundation of America (CCFA). The order set included antibiotics, steroids, deep vein thrombosis prophylaxis, and Clostridium difficile testing. The objective was to improve 1) clinical care and 2) medical knowledge of IBD flares. The study was performed with support from an American College of Gastroenterology Smaller Programs grant (2014). Methods: 368 patients admitted to our institution for IBD exacerbation (269 in the pre-order set group (2010-2012) and 81 in the post-order set group (2014-2015)) were compared in terms of three primary outcomes: chemical deep vein thrombosis (DVT) prophylaxis, Clostridium difficile testing and length of stay. Surgery during admission, immunomodulatory and antibiotic use were also compared. The knowledge gain of first year residents over the year of CPOE roll-out was assessed through the use of surveys collected before and after the completion of their intern year. The 8-question survey was accompanied by monthly short sessions presented to internal medicine house officers during sign-out rounds, reminding them to use the electronic order set and of its clinical relevance. Results: There was a statistically significant improvement in chemical DVT prophylaxis from 62% to 81% (p-value 0.001), decrease in antibiotic use from 100% to 74%(p-value The resident survey demonstrated an 18-point average gain in test scores from 27.8% to 45.9% (p-value 0.001) after the intervention. (Table 2)Figure 1Figure 2Conclusion: This study demonstrates the potential clinical and educational benefits of an electronic order set. We believe it is reproducible in a variety of clinical settings, from Epic-naïve community hospitals such as ours to tertiary care centers. Further work is needed to determine the specific components of the order set as well as of the educational piece that had the highest impact on quality improvement.

Jordan derivations of incidence algebras

Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan derivation of $I(X,\mathcal{R})$ is a derivation provided that $\mathcal{R}$ is $2$-torsion free.

Unifying order structures for Colombeau algebras

We define a general notion of set of indices which, using concepts from pre-ordered sets theory, permits to unify the presentation of several Colombeau-type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau's notion of big-O such that its usual properties continue to hold. Using this generalized notion of big-O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.

Deprecation based greedy strategy for target set selection in large scale social networks

The problem of target set selection for large scale social networks is addressed in the paper. We describe a novel deprecation based greedy strategy to be applied over a pre-ordered (as obtained with any heuristic influence function) set of nodes. The proposed algorithm runs in iteration and has two stages, (i) Estimation: where the performance of each node is evaluated and (ii) Marking: where the nodes to be deprecated in later iterations are marked. We have theoretically proved that for any monotonic and sub-modular influence function, the algorithm correctly identifies the nodes to be deprecated. For any finite set of input nodes it is shown that the algorithm can meet the ending criteria. The worst case performance of the algorithm, both in terms of time and performance, is also analyzed. Experimental results on seven un-weighted as well as weighted social networks show that the proposed strategy improves the ranking of the input seeds in terms of the total number of nodes influenced.

Normalized and strict NaP-preferences

A NaP-preference (necessary and possible preference) is a pair of nested reflexive binary relations having a preorder as its smaller component, and satisfying natural forms of mixed completeness and mixed transitivity. A NaP-preference is normalized if its smaller component is a partial order. Dually, a strict NaP-preference is a pair of nested asymmetric binary relations having a strict partial order as its smaller component and satisfying suitable mixed transitivity properties. We show that normalized and strict NaP-preferences on the same ground set are in a one-to-one correspondence. It is known that a NaP-preference can be characterized by the existence of a set of total preorders whose intersection and union are respectively equal to its two components. In the same spirit, we characterize normalized and strict NaP-preferences by means of suitable families of order relations, respectively called injective and projective. The properties of injectivity and projectivity are a collectionwise extension of the antisymmetry and the completeness of a single binary relation.

ELECTRE-III-H: An outranking-based decision aiding method for hierarchically structured criteria

This paper proposes a method for ranking a set of alternatives evaluated using multiple and conflicting criteria that are organised in a hierarchical structure. The hierarchy permits the decision maker to identify different intermediate sub-problems of interest. In that way, the analysis of the criteria is done according to the subsets defined in the hierarchy, and following the precedence relations in a bottom-up approach. To deal with this type of hierarchical structures, an extension of the ELECTRE-III method, called ELECTRE-III-H, is presented. As all methods of ELECTRE family, this one also relies on building a binary outranking relation on the set of alternatives on the basis of concordance and discordance tests. The exploitation of this outranking relation generates a partial pre-order, establishing an indifference, preference or incomparability relation for each pair of alternatives. The idea of a bottom-up application of the classical ELECTRE-III method to sub-problems involving subsets of criteria at the intermediate levels of the hierarchy is infeasible because the evaluations of alternatives by criteria aggregating some sub-criteria have the form of partial pre-orders, and not complete pre-orders. Thus, we propose a new procedure for building outranking relations from a set of partial pre-orders, as well as a mechanism for propagating these pre-orders upwards in the hierarchy. With this method, the decision maker is able to analyse the problem in a decomposed way and gain information from the outputs obtained at intermediate levels. In addition, ELECTRE-III-H gives the decision maker the possibility to define a local preference model at each node of the hierarchy, according to his objectives and sub-problem characteristics. We show an application of this method to rank websites of tourist destination brands evaluated using a hierarchy with 4 levels.

Gaps in the inpatient management of chronic obstructive pulmonary disease exacerbation and impact of an evidence-based order set.

Evidence-based, guideline-recommended practices improve multiple outcomes in patients admitted with acute exacerbation of chronic obstructive pulmonary disease (AECOPD), but are incompletely implemented in actual practice. Admission order sets with evidence-based diagnostic and therapeutic guidance have enabled quality improvement and guideline implementation in other conditions. To characterize the magnitude of care gaps and the effect of order sets on quality of care in patients with AECOPD. The authors prospectively designed a standardized chart review protocol to document process of care and health care utilization before and after implementation of AECOPD order sets at an academic hospital in Toronto, Ontario. A total of 243 total AECOPD admissions and multiple important care gaps were identified. There were 74 admissions in the pre-order set period (January to June 2009) and 169 in the order set period (October 2009 to September 2010). The order set was used in 78 of 169 (46.2%) admissions. In the order set period, we observed improvements in respiratory therapy educational referrals (five of 74 [6.8%] versus 48 of 169 [28.4%]; P<0.01); venous thromboembolism prophylaxis prescriptions (when indicated) (15 of 68 [22.1%] versus 100 of 134 [74.6%]; P<0.01); systemic steroid prescriptions (55 of 74 (74.3%) versus 151 of 169 [89.4%]; P<0.01]); and appropriate antibiotic prescriptions (nine of 24 [37.5%] versus 61 of 88 [69.3%]; P<0.01). The mean (± SD) length of stay also decreased from 6.5 ± 7.7 days before order sets to 4.1 ± 5.0 days with order sets (P=0.017). Care gaps in inpatient AECOPD management were large and evidence-based order sets may improve guideline adherence at the point of care. Randomized trials including patient outcomes are required to further evaluate this knowledge translation intervention.

Study of Albumin Use in Liver Diseases at a Tertiary Care Center and Effect of a Computerized Order Set

Purpose: Albumin is frequently used for the treatment or prevention of complications of liver cirrhosis. However, albumin is an expensive medication. Albumin use should therefore be evidence-based and appropriate, in order to maximize its value and reduce the risk of complications. The aim was to study the impact of a computerized clinical decision support (CDS) order set on albumin utilization for liver disease in a larger academic tertiary care center. Methods: An appropriate indications list for the use of albumin in liver disease was created based on a systematic review of international guidelines. The appropriate indications for albumin use in liver diseases were deemed to be (1) diagnosis of HRS, (2) treatment of HRS along with vasoconstrictors, (3) large volume paracentesis (>5 litres or Cr >1.5), (4) treatment of SBP only if serum creatinine >1 mg/dL, BUN >30, or total bilirubin >4 mg/dL. A CDS order set was then unveiled throughout the health system that required providers to choose an appropriate indication before order submission. No formal education on appropriate albumin utilization was otherwise given to ordering providers. To assess the impact of the CDS order set alone on albumin utilization in liver disorders, a retrospective chart review of adult in-patients who received albumin 3 months prior to implementation of order set and 3 months post implementation of order set was done. For statistical analysis we used test of the difference in proportions with a p-value of <0.05 considered statistically significant. Results: Pre-order set there were a total of 106 albumin orders (70% appropriate) amounting to 9,498.5 grams of albumin (82% appropriate). Post-order set there were 185 orders (75% appropriate) amounting to 19,295 grams of albumin (84% appropriate). The most common indications for albumin use were large volume paracentesis, diagnosis and treatment of HRS. There was statistically significant improvement in amount of albumin (in grams) used appropriately for large volume paracentesis and treatment of SBP (p<0.001), while there was no difference in number of orders. There was also a statistically significant increase in number of inappropriate orders (p=0.005) and amount of albumin (p<0.001) used in the diagnosis and treatment of HRS. Conclusion: The implementation of a CDS order set without a companion education program decreased inappropriate albumin use for large volume paracentesis and treatment of SBP, however, the inappropriate use of albumin for management of HRS increased. Though the order set had a modest impact, we conclude that institutions should also initiate educational and targeted physician interventions to increase compliance with guidelines and decrease inappropriate albumin use and associated costs.

Upper bounds for continuous seminorms and special properties of bilinear maps

If E is a locally convex topological vector space, let (P(E),≼) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for θ an infinite cardinal those locally convex spaces E which have the θ-neighbourhood property introduced by E. Jordá, meaning that all sets M of continuous seminorms of cardinality |M|⩽θ have an upper bound in P(E). On the other hand, we study bilinear maps β:E1×E2→F between locally convex spaces which admit “product estimates” in the sense that for all pi,j∈P(F), i,j=1,2,… , there exist pi∈P(E1) and qj∈P(E2) such that pi,j(β(x,y))⩽pi(x)qj(y) for all (x,y)∈E1×E2. The relations between these concepts are explored, and examples given. The main applications concern spaces Ccr(M,E) of vector-valued test functions on manifolds.

Evidential reasoning in large partially ordered sets

The Dempster-Shafer theory of belief functions has proved to be a powerful formalism for uncertain reasoning. However, belief functions on a finite frame of discernment Ω are usually defined in the power set 2Ω, resulting in exponential complexity of the operations involved in this framework, such as combination rules. When Ω is linearly ordered, a usual trick is to work only with intervals, which drastically reduces the complexity of calculations. In this paper, we show that this trick can be extrapolated to frames endowed with an arbitrary lattice structure, not necessarily a linear order. This principle makes it possible to apply the Dempster-Shafer framework to very large frames such as the power set, the set of partitions, or the set of preorders of a finite set. Applications to multi-label classification, ensemble clustering and preference aggregation are demonstrated.

New Order Relations in Set Optimization

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set $\mathcal{P}(Y)$ of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set $(\mathcal{Q},\preccurlyeq)$ without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order $\preccurlyeq$ : minimal elements, semicompactness, completeness, domination property of a subset of $\mathcal{Q}$ , and semicontinuity of a set-valued map with values in $\mathcal{Q}$ in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on $(\mathcal{Q},\preccurlyeq)$ from which one can easily derive similar results for the case, when F takes values on $\mathcal{P}(Y)$ equipped with various order relations.

Structured lattices and topological categories of L-sets

Goguen's category of V-sets whose objects are functions with values in a pre-ordered set and morphisms are suitable maps is shown to be a topological construct if and only if the pre-ordered set is a complete lattice; in particular the category Set(L) of L-sets, also considered by Goguen, is topological over Set. The special case when the considered pre-ordered set is L with the “mapping to” relation arising from a structure Φ = ( ϕ a ) a ∈ L , where every ϕ a : L → [ ⊥ , a ] preserves arbitrary infs, already considered by the authors on any complete lattice L, is investigated in detail.

HOMOLOGICAL SYSTEMS IN MODULE CATEGORIES OVER PRE-ORDERED SETS

We introduce the so-called homological systems in a module category over a pre-ordered set, which generalize the notion of a stratifying system over a linearly ordered set, and study both the corresponding modules filtrated by the systems and algebras stratified by the systems. In particular, we recover the tilting theory for pre-standardly stratified algebras, and get a general formula for computing the Cartan determinants of pre-standardly stratified algebras in terms of standard modules and simple modules. Also, the finitistic dimension of a given algebra, and the relative homological dimensions of full subcategories of the modules related to a homological system, are discussed. As an application, we get a new bound for the finitistic dimension of a pre-standardly stratified algebra.

Reasoning with Non-Numeric Linguistic Variables

Where decisions are based on imprecise numeric data and linguistic variables, the development of automated decision aids presents particular difficulties. In such applications, linguistic variables often take their values from a pre-ordered set of vaguely defined linguistic terms. The mathematical structures that arise from the assumption that sets of linguistic terms are pair-wise tolerant are considered. A homomorphism between tolerance spaces, filter bases and fuzzy numbers is shown. A proposal for modeling linguistic terms with an ordered set of fuzzy numbers is introduced. A procedure for structured knowledge acquisition based on the topology of the term sets and the cognitive theory of prototypes is shown to give rise to sparse rule bases. Similarity as a function of “distance” between fuzzy numbers treated as tolerance mappings is used as an inference mechanism in sparse rule bases to give linguistically valued outputs. Measuring the “distance” between fuzzy sets to correspond to intuitive notions of nearness is not straightforward, since the usual metric axioms are not adequate. An alternative way of measuring “distance” between fuzzy numbers is introduced, which reduces to the usual one when applied to crisp numbers.

Groupoidal completely distributive lattices

The Adamek and Pedicchio proof that top op is a quasi-variety is adapted to show that the opposite of the category of pre-ordered sets is also a quasi-variety. The constructive proof given requires a description of power objects in terms of (constructively) completely distributive lattices and such a description is provided by the Carboni and Walters notion of “groupoidal” object in a cartesian bicategory.

Square-free algebras and their automorphism groups

A finite-dimensional algebra A over a field K is square-free in case for every pair e,f of primitive idempotents in Adim K (e A f) ≤ 1. For example, every incidence algebra of a finite pre-ordered set over a field is square-free. The automorphism groups of the latter have been studied by Stanley, Scharlau, Baclawski, and more recently by Coelho. In this paper we characterize all finite-dimensional square-free K-algebras A as certain semigroup algebras A ≅ K ξ S over a square-free semigroup S twisted by some ξ ∈Z 2 (S K *), a two-dimensional cocycle of S with coefficients in the group of units A * of K. We prove that for each such A ≅ K ξ S, its outer automorphism group Out A is the middle term of a short exact sequence where H 1 (S K *) is the first cohomology group of S with coefficients in K *Aut0 S is the group of “normal” automorphisms of the semigroup S, and Stabξ(Auto S) is the stabilizer in Auto S of ξ under the action of Auto S on H 2 (S K * ). Finally, if ξ ≅ 1, so that A ss KS is untwisted, then the sequence splits.

Automorphisms and Higher Derivations of Incidence Algebras

Let I(P, A) be the incidence algebra of a locally finite pre-ordered set P over an algebra A. We derive decompositions of algebra maps I(P, A) → I(P, B) into simpler factors, under sufficient assumptions. As corollaries we deduce decompositions of algebra automorphisms and higher derivations of I(P, A), and we also formulate results in the special cases of upper triangular and full matrix algebras over A. This yields extensions of several older results of a similar type.

Nash Equilibrium for Set-Valued Maps

We define a notion of Nash equilibrium for an arbitrary family of set-valued maps with values in pre-ordered sets. An existence theorem of such a Nash equilibrium is obtained from an intersection theorem for an arbitrary family of set-valued maps, generalizing results of Ky Fan, for example.

Incidence rings of pre-ordered sets.

Incidence rings of pre-ordered sets.