An Overview of the Foundations of the Hypergroup Theory

This paper is written in the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is the prominent structure of hypercompositional algebra. In the beginning, it reveals the structural relation between two fundamental entities of abstract algebra, the group and the hypergroup. Next, it presents the several types of hypergroups, which derive from the enrichment of the hypergroup with additional axioms besides the ones it was initially equipped with, along with their fundamental properties. Furthermore, it analyzes and studies the various subhypergroups that can be defined in hypergroups in combination with their ability to decompose the hypergroups into cosets. The exploration of this far-reaching concept highlights the particularity of the hypergroup theory versus the abstract group theory, and demonstrates the different techniques and special tools that must be developed in order to achieve results on hypercompositional algebra.

Gruppe-ekspertise i pædagogik med henblik på omsorgsarbejde

ResumeArtiklen argumenterer for, at pædagogisk ekspertise kan tilskrives grupper og ikke kun individer. Hvordan skal vi helt fundamentalt forstå pædagogisk gruppe-ekspertise? Artiklen anlægger et pædagogisk-filosofisk perspektiv på dette spørgsmål. Med et fokus på pædagogisk omsorgsarbejde afklarer den, hvad de fundamentale forudsætninger er for gruppeekspertise. Første del undersøger minimumsbetingelserne for, hvad man kan kalde pædagogisk ekspertise. Idealet om pædagogisk ekspertise må forstås bredt, hvis vi skal fastholde det som et meningsfuldt forbillede for pædagogisk teori og praksis. Artiklen analyserer og kategoriserer væsentlige aspekter af ekspertise i en pædagogisk sammenhæng. Pædagogisk ekspertise må rumme en pluralisme af former for viden og kompetencer, både praktiske, teoretiske, før-refleksive og refleksive, samt adfærdsdispositioner, emotionel opmærksomhed og intellektuelle kapaciteter. Artiklens anden del undersøger de filosofiske betingelser for gruppe-ekspertise og drager nogle vigtige distinktioner. Det argumenteres således, at grupper kan have overlegen viden qua gruppe, som de enkelte medlemmer hver især ikke besidder, og denne viden kan under de rette betingelser udgøre ekspertise. For at anskueliggøre denne forståelse af gruppeekspertise illustrerer artiklen til slut idéen i en sammenhæng af pædagogisk omsorgsarbejde. Her eksemplificeres det, hvordan gruppe-ekspertise i pædagogisk omsorgsarbejde kan bestå af samspillet mellem konstitutive delelementer. Abstract Group expertise in pedagogy for care workThe article argues that pedagogical expertise can be attributed to groups and not just individuals. How should we fundamentally understand pedagogical group expertise? The article takes a pedagogical-philosophical perspective on this issue. With a focus on pedagogical care work, it clarifies the fundamental prerequisites for group expertise. The first part examines the minimum conditions for what can be called pedagogical expertise. If we are to maintain it as a meaningful ideal of pedagogical theory and practice, pedagogical expertise must be understood broadly. The article analyzes and categorizes significant aspects of expertise in a pedagogical context. Pedagogical expertise must accommodate a pluralism of knowledge forms and competencies. These include practical, theoretical, pre-reflexive and reflexive competencies, behavioral dispositions, emotional awareness, and intellectual capacities. The second part of the article examines the philosophical conditions of group expertise and draws some important distinctions. Groups can have superior knowledge, which the individual members do not possess individually, and this knowledge can, under the right conditions, constitute expertise. Finally, an example of how constitutive elements of group expertise in pedagogical care work can unfold illustrates the main idea of group expertise in a specific pedagogical care work context.

Invariance groups of functions and related Galois connections

Invariance groups of sets of Boolean functions can be characterized as Galois closures of a suitable Galois connection. We consider such groups in a much more general context using group actions of an abstract group and arbitrary functions instead of Boolean ones. We characterize the Galois closures for both sides of the corresponding Galois connection and apply the results to known group actions.

Elaborating on the Abstract: Group Meaning-Making in a Colombian Microsavings Program

Access to formal financial products like savings accounts constitutes a hallmark feature of economic development, but individuals do not uniformly embrace these products. In explaining such financial preferences, scholars have focused on institutional, cultural, and material factors, but they have paid less attention to organizations and small groups. In this article, we argue that these factors are crucial to understanding financial preferences. We investigate a government-sponsored microsavings program in Colombia and find that participants became less interested in banking services over the course of the program, even as they gained access to appropriate accounts and their savings increased. Turning to qualitative data to understand this curious finding, we show that organizational efforts to disseminate abstract information about banking triggered a process of “elaboration” among group members, leading many to develop financial preferences at odds with those promoted by the government. This study integrates insights from economic sociology, organizational theory, and microsociology to advance theories of financial preference. In doing so, we reveal how organizational efforts to compress information, followed by group efforts to personalize and expand upon the information, can shape preferences and potentially undermine organizational goals.

HOW TO BEST FRAME A DESIGN BRIEF TO MAXIMIZE NOVELTY AND USEFULNESS IN IDEA GENERATION

This paper aims to identify factors that influence creativity, and strives towards understanding the effect of representations, namely abstract and concrete design outcomes. Three conditions are compared; a control group, an abstract group, and a group provided with various example solutions. The implications of this work can strongly impact the formulation of design briefs, where the goal is to stimulate the creativity of design brief outcomes and examine their relationship to product awareness.

Dimension groups for self-similar maps and matrix representations of the cores of the associated C*-algebras

We introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -group of the core of the C*-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ together with the unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

Specificity of genetic nature and classification characteristics of Zmiiny Island chernozem soils

Even the first researchers of the nature of the small (20.5 hectares) rocky Zmiiny Island in the northwestern Black Sea region visually diagnosed the soil there with black earth. Since 2003, the staff of the Department of Soil Science and Soil Geography of the Odessa National University named by I.U.Mechnikov ONU began the study of factors and processes of soil formation, soils and soil cover of Zmiiny Island territory. It was found out that on the inter-rocky areas of the island un- der the steppe grassy vegetation on the gravelly-stony eluvium or eluvium-deluvium of dense rocks formed unusually high humus (up to 14-18%) undeveloped and short-profile chernozem soils. However, the diagnostic and classification of these soils remains unclear. The purpose is to characterize the specificity of the genetic nature of the island soils and to determine their diagnostic and classification affiliation, including the World Reference Base for Soil Resources WRB.The materials presented in this article are obtained as a result of our extensive fieldwork and laboratory analysis, research and mapping of soils and soil cover of Zmiiny Island with the application of methodological principles of the process-genetic paradigm.Chernozem soils on the island are formed on a low-strength (25-40, up to 50 cm) rocky-gravelly crust of weathering of dense acidic rocks under the cover of almost virgin steppe herbaceous vegetation and year-round salting from the sea, mainly chlorides and Na sulfates. The soils are short-profile, typical chernozem humus-accumulative type with typical for the chernozem profile genetic horizons, gravel-stony, non-carbonate, almost non-structural, acidic, varyingly saline and solonetinous, but without solonets-ileuvial profile differentiation. The humus content in the upper horizons is very high (10-12, up to 15-18%), it is typical of the chernozem composition of the Chumic acid : Сfulvic acid (2,5-3,3), but with a sharp dominance of the HA-1 fraction (70% and more) and very low content (1-2, up to 3-4%) or lack of typical chernozem HA-2fraction. It is established that there is a tendency to an irreversible constant increase of the chernozem properties and characteristics in soils as the bioproductivity of herbal vegetation increases.For the first time their diagnostic and classification dependence on the system of both domestic and World WRB soil classifications has been determined. The WRB system proposes to refer these soils to the Phaeozem abstract group, with clarification of the classifiers Leptic and Skeletic. The work is practically significant in terms of the integration the Ukrainian soil science in the system of the current international standard of soil classification WRB.