Normalish Amenable Subgroups of the R. Thompson Groups

Results in [Formula: see text] algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups [Formula: see text]. These results together show that [Formula: see text] is non-amenable if and only if [Formula: see text] has a simple reduced [Formula: see text]-algebra. In further investigations into the structure of [Formula: see text]-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group [Formula: see text]. They show that if a group [Formula: see text] admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced [Formula: see text]-algebra. Our chief result concerns the R. Thompson groups [Formula: see text]; we show that there is an elementary amenable group [Formula: see text] [where here, [Formula: see text]] with [Formula: see text] normalish in [Formula: see text]. The proof given uses a natural partial action of the group [Formula: see text] on a regular language determined by a synchronising automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of [Formula: see text] with various forms of formal language theory.

Partial actions of weak Hopf algebras on coalgebras

In this work, the notions of a partial action of a weak Hopf algebra on a coalgebra and of a partial action of a groupoid on a coalgebra will be introduced, together with some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of a partial action on a coalgebra and of a partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.

On partial skew groupoid rings

Given a partial action [Formula: see text] of a connected groupoid [Formula: see text] on an associative ring [Formula: see text] we investigate under what conditions the partial skew groupoid ring [Formula: see text] can be realized as a partial skew group ring. In such a case, applications concerning to the separability, semisimplicity and Frobenius property of the ring extension [Formula: see text] as well as to the artinianity of [Formula: see text] are given.

Hopf algebras arising from partial (co)actions

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1) (1990) 17–64.], we introduce the notion of partial matched pair [Formula: see text] involving the concepts of partial action and partial coaction between two bialgebras [Formula: see text] and [Formula: see text]. Furthermore, we present sufficient conditions for the corresponding bismash product [Formula: see text] to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.

Galois corings and groupoids acting partially on algebras

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

An Assimilation of TRIZ in Dissecting the Statistical Outcomes of Tactile Sensitivity, Pinch Force and Endurance among Elderly People

Background: Numerous investigations have analysed the handgrip force and endurance of elderly people. However, few studies examine reduced and increased tactile sensibility effects on the pinch force and endurance of elderly people. Even fewer studies include the problem-solving process extending from statistical outcomes of such studies. This study examines tactile sensibility effects on the pinch force and endurance of elderly people, and potentially resolves issues dissected from these inferences using TRIZ. Methods: Data on pinch force and endurance time was collected among 32 subjects aged 55-65 years old. Subjects were required to pinch an experimental apparatus at their maximum limit, and sustain their pinching activity for as long as possible. Cotton gloves were for reduced tactile sensibility while rubber gloves were for increased tactile sensibility. The two-sample T-test results were further analysed using TRIZ.Results: The results suggested that a significant difference existed between the pinch force from reduced and increased tactile sensibility (p < 0.05), with similar outcomes for endurance time. Resolving TRIZ contradictions identified from the results presented this study with a principle known as “partial action”, which suggested that elderly people should pinch using less of the originally desired force when the exact intended force is difficult to achieve, rather than exerting a high pinch force in a single attempt. The “segmentation” and “other way around” principles were also recommended. Through Su-Field analysis, it was found that harmful effects from pinching can be neutralised using intermediary materials between the fingers and object, such as rubber. The analysis also proposed using optical or acoustic fields, whereby light sensors or buzzers could act as mechanisms to provide signals once a sufficient pinch force is detected. Conclusion: This study confirmed that elderlies with poor tactile sensibility commonly pinch objects with excessive force and extended durations. The inventive solutions that extend from this finding through TRIZ provide new insights to researchers in product design with the aim of resolving poor pinch performance caused by degrading tactile sensibility.

Multiplier Hopf algebras: Globalization for partial actions

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

Two-sided expansions of monoids

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid [Formula: see text] and a class of partial actions of [Formula: see text], determined by a set, [Formula: see text], of identities, we define [Formula: see text] to be the universal [Formula: see text]-generated two-sided restriction monoid with respect to partial actions of [Formula: see text] determined by [Formula: see text]. This is an [Formula: see text]-restriction monoid which (for a certain [Formula: see text]) generalizes the Birget–Rhodes prefix expansion [Formula: see text] of a group [Formula: see text]. Our main result provides a coordinatization of [Formula: see text] via a partial action product of the idempotent semilattice [Formula: see text] of a similarly defined inverse monoid, partially acted upon by [Formula: see text]. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of [Formula: see text] agree well with suitable properties of [Formula: see text], such as being cancellative or embeddable into a group. We observe that if [Formula: see text] is an inverse monoid, then [Formula: see text], the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion [Formula: see text]. This gives a presentation of [Formula: see text] and leads to a model for [Formula: see text] in terms of the known model for [Formula: see text].

Goldie twisted partial skew power series rings

In this paper, we work with a unital twisted partial action of [Formula: see text] on a unital ring [Formula: see text]. We introduce the twisted partial skew power series rings and twisted partial skew Laurent series rings. We study primality, semi-primality and the prime ideals in these rings. We describe the prime radical in twisted partial skew Laurent series rings. We investigate the Goldie property in twisted partial skew power series rings and twisted partial skew Laurent series rings. Moreover, we describe conditions for the semiprimality in twisted partial skew power series rings.