Idempotent set-theoretical solutions of the pentagon equation

A set-theoretical solution of the pentagon equation on a non-empty set $X$ is a function $s:X\times X\to X\times X$ satisfying the relation $s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$, with $s_{12}=s\times \,id_X$, $s_{23}=id_X \times \, s$ and $s_{13}=(id_X\times \, \tau)s_{12}(id_X\times \,\tau)$, where $\tau:X\times X\to X\times X$ is the flip map given by $\tau(x,y)=(y,x)$, for all $x,y\in X$. Writing a solution as $s(x,y)=(xy ,\theta_x(y))$, where $\theta_x: X \to X$ is a map, for every $x\in X$, one has that $X$ is a semigroup. In this paper, we study idempotent solutions, i.e., $s^2=s$, by showing that the idempotents of $X$ have a key role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $\theta_1$ is a monoid homomorphism

Solutions of the Yang-Baxter equation and strong semilattices of skew braces

We prove that any set-theoretic solution of the Yang-Baxter equation associated to a dual weak brace is a strong semilattice of non-degenerate bijective solutions. This fact makes use of the description of any dual weak brace $S$ we provide in terms of strong semilattice $Y$ of skew braces $B_\alpha$, with $\alpha \in Y$. Additionally, we describe the ideals of $S$ and study its nilpotency by correlating it to that of each skew brace $B_\alpha$

Deformed solutions of the Yang-Baxter equation coming from dual weak braces and unital near-trusses

We show that a family of set-theoretic solutions comes from dual weak braces, including the recent ones provided in the context of skew braces. In particular, the solutions we give are obtained by ``deforming" the classical ones by certain parameters $z$. In any dual weak brace $(S, +, \circ)$, the parameters giving rise to deformed solutions are only those belonging to the right distributor of $S$, i.e., $\mathcal{D}_r(S)=\{z \in S \, \mid \, \forall \, a,b \in S \quad (a+b) \circ z=a\circ z-z+b \circ z\}$, that is a full inverse subsemigroup of $\left(S, \circ\right)$. Particular attention is posed to the class of skew braces and, in a natural way, of two-sided skew braces. In addition, we construct new arbitrary solutions on the algebraic structure of the unital near-truss determined by extending deformed solutions coming from skew braces

Set-theoretical solutions of the Yang-Baxter and pentagon equations on semigroups

The Yang-Baxter and pentagon equations are two well-known equations of Mathematical Physic. If $S$ is a set, a map $s:S\times S\to S\times S$ is said to be a set theoretical solution of the Yang-Baxter equation if $$s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23},$$ where $s_{12}=s\times id_S$, $s_{23}=id_S\times s$, and $s_{13}=(id_S\times \tau)\,s_{12}\,(id_S\times \tau)$ and $\tau$ is the flip map, i.e., the map on $S\times S$ given by $\tau(x,y)=(y,x)$. Instead, $s$ is called a set-theoretical solution of the pentagon equation if $$s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}.$$ The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang-Baxter equation. Specifically, we present a new construction of solutions of the Yang-Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product

Inverse semi-braces and the Yang-Baxter equation

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple $(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} + c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S, \cdot)$. In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known.Comment: 43 page

Set-theoretical solutions of the pentagon equation on groups

Let $M$ be a set. A set-theoretical solution of the pentagon equation on $M$ is a map $s:M\times M\longrightarrow M\times M$ such that \begin{equation*} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}, \end{equation*} where $s_{12}=s\times id_M$, $s_{23}=id_M \times s$ and $s_{13}=(id_M \times \tau) s_{12}(id_M \times \tau)$, and $\tau$ is the flip map, i.e., the permutation on $M\times M$ given by $\tau(x,y)=(y,x)$, for all $x,y\in M$. In this paper we give a complete description of the set-theoretical solutions of the form $s(x,y)=(x\cdot y , x\ast y)$ when either $(M,\cdot)$ or $(M,\ast)$ is a group; moreover, we raise some questions.Comment: 12 page

Psychometric Properties of the Italian Version of the Leader Member Exchange Scale (LMX-7): A Validation Study

: For decades, scholars have studied leader-member exchange (LMX) relationships to understand and explain the effects of leadership on follower attitudes and performance outcomes within work settings. One available instrument to measure these aspects is the LMX-7 scale. This measurement has been widely used in empirical studies, but its psychometric properties have been poorly explored. The aim of this study was to test the psychometric characteristics (content, structural and construct validity, and reliability) of the Italian version of the LMX-7 scale and to support its cultural adaptation. We used a cross-sectional multi-center design. The forward-backward translation process was used to develop the Italian version of the scale. The scale was administered through an online survey to 837 nurses and nurse managers working in different settings. The factorial structure was tested using both exploratory and confirmatory factor analyses (EFA and CFA), and reliability was evaluated using Cronbach's alpha. For the construct validity, we used hypothesis testing and differentiation by known groups. The Italian version of the LMX-7 scale presented one dimension. All the psychometric tests performed confirmed its validity and suggested its usefulness for future research

Effects of pre‐operative isolation on postoperative pulmonary complications after elective surgery: an international prospective cohort study

We aimed to determine the impact of pre-operative isolation on postoperative pulmonary complications after elective surgery during the global SARS-CoV-2 pandemic. We performed an international prospective cohort study including patients undergoing elective surgery in October 2020. Isolation was defined as the period before surgery during which patients did not leave their house or receive visitors from outside their household. The primary outcome was postoperative pulmonary complications, adjusted in multivariable models for measured confounders. Pre-defined sub-group analyses were performed for the primary outcome. A total of 96,454 patients from 114 countries were included and overall, 26,948 (27.9%) patients isolated before surgery. Postoperative pulmonary complications were recorded in 1947 (2.0%) patients of which 227 (11.7%) were associated with SARS-CoV-2 infection. Patients who isolated pre-operatively were older, had more respiratory comorbidities and were more commonly from areas of high SARS-CoV-2 incidence and high-income countries. Although the overall rates of postoperative pulmonary complications were similar in those that isolated and those that did not (2.1% vs 2.0%, respectively), isolation was associated with higher rates of postoperative pulmonary complications after adjustment (adjusted OR 1.20, 95%CI 1.05-1.36, p = 0.005). Sensitivity analyses revealed no further differences when patients were categorised by: pre-operative testing; use of COVID-19-free pathways; or community SARS-CoV-2 prevalence. The rate of postoperative pulmonary complications increased with periods of isolation longer than 3 days, with an OR (95%CI) at 4-7 days or >= 8 days of 1.25 (1.04-1.48), p = 0.015 and 1.31 (1.11-1.55), p = 0.001, respectively. Isolation before elective surgery might be associated with a small but clinically important increased risk of postoperative pulmonary complications. Longer periods of isolation showed no reduction in the risk of postoperative pulmonary complications. These findings have significant implications for global provision of elective surgical care

Screening policies, preventive measures and in-hospital infection of COVID-19 in global surgical practices

Background In a surgical setting, COVID-19 patients may trigger in-hospital outbreaks and have worse postoperative outcomes. Despite these risks, there have been no consistent statements on surgical guidelines regarding the perioperative screening or management of COVID-19 patients, and we do not have objective global data that describe the current conditions surrounding this issue. This study aimed to clarify the current global surgical practice including COVID-19 screening, preventive measures and in-hospital infection under the COVID-19 pandemic, and to clarify the international gaps on infection control policies among countries worldwide.Methods During April 2-8, 2020, a cross-sectional online survey on surgical practice was distributed to surgeons worldwide through international surgical societies, social media and personal contacts. Main outcome and measures included preventive measures and screening policies of COVID-19 in surgical practice and centers' experiences of in-hospital COVID-19 infection. Data were analyzed by country's cumulative deaths number by April 8, 2020 (high risk, >5000; intermediate risk, 100-5000; low risk, <100).Results A total of 936 centers in 71 countries responded to the survey (high risk, 330 centers; intermediate risk, 242 centers; low risk, 364 centers). In the majority (71.9%) of the centers, local guidelines recommended preoperative testing based on symptoms or suspicious radiologic findings. Universal testing for every surgical patient was recommended in only 18.4% of the centers. In-hospital COVID-19 infection was reported from 31.5% of the centers, with higher rates in higher risk countries (high risk, 53.6%; intermediate risk, 26.4%; low risk, 14.8%; P<0.001). Of the 295 centers that experienced in-hospital COVID-19 infection, 122 (41.4%) failed to trace it and 58 (19.7%) reported the infection originating from asymptomatic patients/staff members. Higher risk countries adopted more preventive measures including universal testing, routine testing of hospital staff and use of dedicated personal protective equipment in operation theatres, but there were remarkable discrepancies across the countries.Conclusions This large international survey captured the global surgical practice under the COVID-19 pandemic and highlighted the insufficient preoperative screening of COVID-19 in the current surgical practice. More intensive screening programs will be necessary particularly in severely affected countries/institutions

Evaluation of a quality improvement intervention to reduce anastomotic leak following right colectomy (EAGLE): pragmatic, batched stepped-wedge, cluster-randomized trial in 64 countries

Background Anastomotic leak affects 8 per cent of patients after right colectomy with a 10-fold increased risk of postoperative death. The EAGLE study aimed to develop and test whether an international, standardized quality improvement intervention could reduce anastomotic leaks. Methods The internationally intended protocol, iteratively co-developed by a multistage Delphi process, comprised an online educational module introducing risk stratification, an intraoperative checklist, and harmonized surgical techniques. Clusters (hospital teams) were randomized to one of three arms with varied sequences of intervention/data collection by a derived stepped-wedge batch design (at least 18 hospital teams per batch). Patients were blinded to the study allocation. Low- and middle-income country enrolment was encouraged. The primary outcome (assessed by intention to treat) was anastomotic leak rate, and subgroup analyses by module completion (at least 80 per cent of surgeons, high engagement; less than 50 per cent, low engagement) were preplanned. Results A total 355 hospital teams registered, with 332 from 64 countries (39.2 per cent low and middle income) included in the final analysis. The online modules were completed by half of the surgeons (2143 of 4411). The primary analysis included 3039 of the 3268 patients recruited (206 patients had no anastomosis and 23 were lost to follow-up), with anastomotic leaks arising before and after the intervention in 10.1 and 9.6 per cent respectively (adjusted OR 0.87, 95 per cent c.i. 0.59 to 1.30; P = 0.498). The proportion of surgeons completing the educational modules was an influence: the leak rate decreased from 12.2 per cent (61 of 500) before intervention to 5.1 per cent (24 of 473) after intervention in high-engagement centres (adjusted OR 0.36, 0.20 to 0.64; P < 0.001), but this was not observed in low-engagement hospitals (8.3 per cent (59 of 714) and 13.8 per cent (61 of 443) respectively; adjusted OR 2.09, 1.31 to 3.31). Conclusion Completion of globally available digital training by engaged teams can alter anastomotic leak rates. Registration number: NCT04270721 (http://www.clinicaltrials.gov)