A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.ArticleJOURNAL OF COMBINATORIAL THEORY SERIES A. 117(8):1207-1217 (2010)journal articl

Association schemoids and their categories

We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in the category algebra of the underlying category of a schemoid. In this paper, the equivalence between the categories of grouopids and that of thin association schemoids is established. Moreover linear extensions of schemoids are considered. A general theory of the Baues-Wirsching cohomology deduces a classification theorem for such extensions of a schemoid. We also introduce two relevant categories of schemoids into which the categories of schemes due to Hanaki and due to French are embedded, respectively.Comment: 27 pages, Lemma 6.10 is revised adding a conditio

Association schemoids and their categories

We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in the category algebra of the underlying category of a schemoid. In this paper, the equivalence between the categories of groupoids and that of thin association schemoids is established. Moreover linear extensions of schemoids are considered. A general theory of the Baues-Wirsching cohomology deduces a classification theorem for such extensions of a schemoid. We also introduce two relevant categories of schemoids into which the categories of schemes due to Hanaki and due to French are embedded, respectively.ArticleAPPLIED CATEGORICAL STRUCTURES. 23(2):107-136 (2013)journal articl

Frobenius-Schur indicator for categories with duality

We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.Comment: 38 pages; final version published in the Special Issue on "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" of Axiom

Decomposition algebras and axial algebras

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.Comment: 23 page

Cross-border purchases of health services : a case study on Austria and Hungary

This paper explores the structure of cross-border health purchasing between Austria and Hungary and determines the size of this phenomenon as well as the barriers to a further increase. Austrian patients may receive health care treatment in Hungary in three different ways. First, patients may receive benefits in the context of the European Community Regulations 1408/71 and 574/72 (Category I patients). Second, outside those regulatory structures, Austrian patients travel to Hungary to receive medical treatment, especially dental treatment, and then seek reimbursement from their Austrian insurance (Category II patients). Third, some patients receive medical treatment in Hungary outside both schemes (Category III patients). There are about 42,500 Category I patients per year; and 58,000 Category II patients world-wide per year. An unknown but supposedly greater number of patients travel to Hungary to receive mainly dental treatment and cosmetic surgery (Category III). Most health actors in both Austria and Hungary do not regard cross-border purchasing of health services as having cost-saving effects. They put forward major legal, institutional, political, and psychological barriers, which inhibit public and private Austrian providers, to facilitate trade in health care and which inhibit individual patients to realize cost savings through capitalizing on lower health care prices in Hungary. Therefore, for the time being, trade in health care and patient mobility between Austria and Hungary is a circumscribed phenomenon in terms of quantities, and it will most probably remain so in the near future.access to health care; adequate resources; aid; beds; cataract surgery; clinics; Community hospitals; Consumer Protection; cost effectiveness; costs of treatment; dental care; dental treatment; dentists; Diagnosis; discrimination; disease; doctor; doctors; domestic law; employment; entitlement; expenditures; families; financial resources; fundamental principles; general practitioner; Health Affairs; health care; health care centers; health care costs; health care coverage; health care facilities; health care institutions; health care insurance; health care law; health care provider; health care providers; health care sector; health care services; health care standards; health care system; health care systems; Health Care Systems in Transition; health expenditure; health facilities; health insurance; health insurance companies; health insurance funds; health insurance system; health insurers; Health Organization; health organizations; health policy; health providers; health sector; health service; Health Services; health system; health systems; Health Systems in Transition; Healthcare; hospital care; hospital financing; Hospital Operator; hospital sector; hospital treatment; hospitals; hygiene; income; insurance; insurance coverage; insurance systems; Integration; judicial proceedings; legal provisions; marketing; Medical Association; medical associations; medical benefits; medical care; medical facilities; medical science; medical services; medical treatment; medicine; Migration; National Health; National Health Insurance; National Health Insurance Fund; national health policy; nurses; patient; patient care; patient treatment; patients; physician; physicians; Policy ReseaRch; Primary Care; private health insurance; private health insurers; private hospitals; private households; private insurance; private insurer; private insurers; private sector; provision of health care; provision of services; public health; public health care; public health insurance; public hospitals; public sector; quality control; quality of health; quality of health care; rehabilitation; reimbursement rates; right to health care; social health insurance; social insurance; Social Policy; social security; social security schemes; social security systems; surgery; therapy; treatments; Use of Health Care Services; visits; workers

On strong homotopy for quasi-schemoids

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. As a homotopy invariant, the homotopy set of self-homotopy equivalences on a quasi-schemoid is introduced. The main theorem enables us to deduce that the homotopy invariant for the quasi-schemoid induced by a finite group is isomorphic to the automorphism group of the given group.Comment: 12 page