Errata to the paper "On exact sequences and strict bounded group homomorphisms "

Corrections to the above-mentioned paper are provided.

Топологические решеточно упорядоченные кольца с AM-свойством

Motivated by the recent definition of the $AM$-property in locally solid vector lattices [O. Zabeti, doi: 10.1007/s41980-020-00458-7], in this note, we try to investigate some counterparts of those results in the category of all locally solid lattice rings. In fact, we characterize locally solid lattice rings in which order bounded sets and bounded sets agree. Furthermore, with the aid of the $AM$-property, we find conditions under which order bounded group homomorphisms and different types of bounded group homomorphisms coincide. Moreover, we show that each class of bounded order bounded group homomorphisms on a locally solid lattice ring $X$ has the Lebesgue or the Levi property if and only if so is $X$.

On exact sequences and strict bounded group homomorphisms

Necessary and sufficient conditions for the exactness (in the algebraic sense) of certain sequences of bounded group homomorphisms are established.

Low-dimensional bounded cohomology and extensions of groups

Bounded cohomology of groups was first studied by Gromov in 1982 in his seminal paper M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic “non-positively curved” groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension $2$ and $3$ in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by K. Fujiwara and M. Kapovich, On quasihomomorphisms with noncommutative targets, Geom. Funct. Anal. 26 (2016), no. 2, 478–519.

A few remarks on bounded homomorphisms acting on topological lattice groups and topological rings

Suppose G is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on G which have topological structures. In this paper, our attempt is to assign lattice structures on them. More precisely, we use of a version of the remarkable Riesz-Kantorovich formulae and Fatou property for bounded order bounded homomorphisms to allocate the desired structures. Moreover, we show that unbounded convergence on a locally solid lattice group is topological and we investigate some applications of it. Also, some necessary and sufficient conditions for completeness of different types of bounded group homomorphisms between topological rings have been obtained, as well.

Решеточная структура в пространстве ограниченных гомоморфизмов между топологическими решеточно упорядоченными кольцами

Suppose X is a topological ring. It is known that there are three classes of bounded group homomorphisms on X whose topological structures make them again topological rings. First, we show that if X is a Hausdorff topological ring, then so are these classes of bounded group homomorphisms on X. Now, assume that X is a locally solid lattice ring. In this paper, our aim is to consider lattice structure on these classes of bounded group homomorphisms more precisely, we show that, under some mild assumptions, they are locally solid lattice rings. In fact, we consider bounded order bounded homomorphisms on X. Then we show that under the assumed topology, they form locally solid lattice rings. For this reason, we need a version of the remarkable RieszKantorovich formulae for order bounded operators in Riesz spaces in terms of order bounded homomorphisms on topological lattice groups.

Handbook of Culture and Memory

This book is about the ways in which culture matters to memory. It explores how memory is deeply entwined with social relationships, stories in film and literature, group history, monuments, ritual practices, material artifacts, and a host of other cultural devices. Culture in this account is not a bounded group of people or variable to be manipulated but, rather, the medium through which people live and make meaning of their lives. The focus of analysis becomes one of understanding the mutual constitution of people’s memories and the social–cultural worlds to which they belong. An interdisciplinary team of leading scholars has been brought together in this volume to offer new theoretical models of memory as both a psychological and a social–cultural process. The following themes are explored: the concept of memory and its relation to evolution, neurology, culture, and history; the particular dynamics of different cultural contexts of remembering, such as families, commemorations, giving testimony, and struggling with difficult memories such as in therapy; life course changes in memory from its development in childhood, through its anticipatory function in emerging adulthood, to managing its decline in old age; and the national and transnational organization of collective memory and identity through narratives propagated in political discourse, the classroom, and media. This book is essential reading for anyone interested in the complex and interconnected relationship between culture, mind, and memory.

Defining “Community” in Community Health Evaluation

This article examines the theoretical basis of the community as it is evoked in health evaluation. In particular, we examine how hospitals carrying out Community Health Needs Assessments (CHNAs) define communities as well as the implications for these definitions for how to study and engage community problems. We present qualitative findings from a sample of Appalachian nonprofit hospitals, who we asked to describe their approach to defining the community in their most recent Internal Revenue Service–mandated CHNA. Drawing upon a theoretical debate in the history of evaluation research, the authors argue that the contemporary community cannot be circumscribed merely by geographic boundaries, nor can it be identified easily with a bounded group of clearly demarcated individuals. Instead, following the tenets of community-based health research, the authors argue for a richer, more dynamic conceptualization of the community in evaluation research in which definitions arise from community bodies themselves.