Extensions of recent combinatorial refinements of discrete and integral Jensen inequalities

The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely f-divergences in information theory (an interesting refinement of the weighted geometric mean–arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder’s and Minkowski’s inequalities.

Essential extensions of filters in residuated lattices

We define the essential extension of a filter in the residuated lattice A associated to an ideal of L(A) and investigate its related properties. We prove the residuated lattice A is a Boolean algebra, G(RL)-algebra or MV -algebra if and only if the essential extension of {1} associated to A \ P is a Boolean filter, G-filter or MV -filter (for all P ∈ SpecA), respectively. Also, some properties of lattice of essential extensions are studied.

The nonpharmaceutical interventionist (NPI) signs of the coronavirus pandemic: a documentary typology and case study of COVID-19 signage

PurposeSigns saturate and surround society. This article illuminates the significant roles played by documentation within the context of the coronavirus pandemic. It centres, what it terms as, “COVID-19 signage” as essential extensions of nonpharmaceutical interventions (NPIs) into society. It posits that this signage helps materialize, mediate and articulate the pandemic from an unseen phenomenon into tangible objects with which people see and interact.Design/methodology/approachThis article presents a documentary typology of COVID-19 signage to provide a conceptual framework in which to situate, approach and analyse this diverse documentation and its implications for social life and traffic. Further, this article offers a case study of Malta's COVID-19 signage that helped materialize, mediate and articulate the pandemic across the European island nation during its national lockdown in the first half of 2020. This case study helps contextualize these signs and serves as a dual contemporary and historical overview of their creation, implementation and use.FindingsThe coronavirus pandemic cannot be seen with the naked eye. It is, in many respects, an abstraction. Documents enable the virus to be seen and the pandemic to be an experienced reality. Specifically, COVID-19 signage materializes the disease and pandemic into tangible items that individuals interact with and see on a daily basis as they navigate society. From personal to environmental to community signs, these documents have come to mediate social life and articulate COVID-19 during this extraordinary health crisis. A material basis of a shared “pandemic social culture” is consequently established by and through this signage and its ubiquity.Research limitations/implicationsThis article can serve as a point of departure for analyses of other kinds of COVID-19 signage in various contexts. It can serve as an anchor or example for other investigations into what other signs were used, including why, when and how they were produced, designed, formatted, implemented, enforced, altered and/or removed. For instance, it could be used for comparative studies between different NPIs and their associated signage, or of the signage appearing between different cities or countries or even the differences in signage at various political and socio-temporal points of the pandemic.Social implicationsIt is dually hoped that this article's documentary typology, and historical snapshot, of COVID-19 signage could help inform how current and future NPIs into society are or can be used to mitigate the coronavirus or other potential health crises as well as serve as both a contemporary and historical snapshot of some of the immediate and early responses to the pandemic.Originality/valueThis documentary typology can be applied to approaches and analyses of other kinds of COVID-19 signage and related documentation. By serving as a conceptual framework in which situate, approach and analyse these documents, it is hoped that this article can help create a sense of clarity in reflections on sign-saturated environments as well as be practically employed for examining and understanding the effective implementation of NPIs in this pandemic and other health crises.

Down closed injectivity and essentialness

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for [Formula: see text]-posets, posets with an action of a pomonoid [Formula: see text] on them. They gave a criterion for dc-injectivity and studied such injectivity for [Formula: see text] itself, and for its poideals. In this paper, we give results about dc-injectivity of [Formula: see text]-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of [Formula: see text]-injectivity for [Formula: see text]-posets, where [Formula: see text] and [Formula: see text] is externally adjoined to the posemigroup [Formula: see text], we find some classes of pomonoids such that for [Formula: see text]-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of [Formula: see text]-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of [Formula: see text]-posets for some classes of pomonoids [Formula: see text].

An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms

Departing from a general definition of injectivity of modules with respect to suitable algebraic classes of morphisms, we establish conditions under which two modules are isomorphic when they are isomorphic to submodules of each other. The main result of this work extends both Bumby’s criterion for the isomorphism of injective modules and the well-known Cantor–Bernstein–Schröder’s theorem on the cardinality of sets. In the way, various properties on essential extensions, injective modules and injective hulls are generalized. The applicability of our main theorem embraces the cases of [Formula: see text]-injective and pure-injective modules as particular scenarios. Many of the propositions which lead to the proof of the main result of this paper are valid for arbitrary categories.