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

Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang-Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions

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

Distributed Logic Objects: A Fragment of Rewriting Logic and its Implementation

Abstract This paper presents a logic language (called Distributed Logic Objects, DLO for short) that supports objects, messages and inheritance. The operational semantics of the language is given in terms of rewriting rules acting upon the (possibly distributed) state of the system. In this sense, the logic underlying the language is Rewriting Logic. In the paper we discuss the implementation of this language on distributed memory MIMD architectures, and we describe the advantages achieved in terms of flexibility, scalability and load balancing. In more detail, the implementation is obtained by translating logic objects into a concurrent logic language based on multi-head clauses, taking advantage from its distributed implementation on a massively parallel architecture. In the underlying implementation, objects are clusters of processes, objects' state is represented by logical variables, message-passing communication between objects is performed via multi-head clauses, and inheritance is mapped into clause union. Some interesting features such as transparent object migration and intensional messages are easily achieved thanks to the underlying support. In the paper, we also sketch a (direct) distributed implementation supporting the indexing of clauses for single-named methods

Pertussis in infants and the resurgence of a vaccine preventable disease: what to do?

Pertussis or whooping cough remains one of the most poorly controlled vaccine-preventable diseases across the world. Universal vaccination has dramatically reduced its incidence but has failed to bring it completely under control. In the last decades, changes in pertussis epidemiology have been noted, likely related to the introduction of acellular pertussis vaccines. Increasing incidence is recorded among adolescents and adults who have become a reservoir for transmission to unimmunized infants, who are at risk of severe disease and death. In Italy, experimental evidences suggest a sustained circulation of Bordetella pertussis in the adult population and a significant health burden of pertussis among infants less than six months of age. Public health systems are currently exploring new vaccination strategies, including a cocooning strategy to prevent the transmission of the disease from family members to the newborn and vaccination of pregnant mothers to transmit protective antibodies to the offspring, and neonatal vaccination. An integrated approach for pertussis control and prevention need to enhance the current surveillance system providing an accurate estimate of the real burden of pertussis in our Country, particularly among infants

Contrasting the anti-vaccine prejudice: a public health perspective

Although immunization is one of the most successful and cost-effective healthinterventions, there has been always opposition to vaccines. This may be due to several factors, some of which are : 1) the vaccines are given to healthy individuals to prevent disease; 2) the perception of the vaccine value paradoxically declines when the use of a vaccine reduces or eliminates the risk of a disease. Contrasting anti-vaccine movements/feelings is important in order to keep vaccinate coverage rates high. Specific training of health care workers and other vaccine providers is needed in order to understand the reasons of reluctant parents, and to deal with prejudice and misinformation

Vaccine coverage and determinants of incomplete vaccination in children aged 12-23 months in dschang, west region, cameroon: a cross-sectional survey during a polio outbreak

Inadequate immunization coverage with increased risk of vaccine preventable diseases outbreaksremains a problem in Africa. Moreover, different factors contribute to incomplete vaccination status. This study wasperformed in Dschang (West Region, Cameroon), during the polio outbreak occurred in October 2013, in order toestimate the immunization coverage among children aged 12–23 months, to identify determinants for incompletevaccination status and to assess the risk of poliovirus spread in the study population.Methods:A cross-sectional household survey was conducted in November-December 2013, using the WHOtwo-stage sampling design. An interviewer-administered questionnaire was used to obtain information fromconsenting parents of children aged 12–23 months. Vaccination coverage was assessed by vaccination card andparents’recall. Chi-square test and multilevel logistic regression model were used to identify the determinants ofincomplete immunization status. Statistical significance was set atp90 %, and 73.4 % children completedthe recommended vaccinations before 1-year of age. In the final multilevel logistic regression model, factorssignificantly associated with incomplete immunization status were: retention of immunization card (AOR: 7.89;95 % CI: 1.08–57.37), lower mothers’utilization of antenatal care (ANC) services (AOR:1.25; 95 % CI: 1.07–63.75),being the≥3rdborn child in the family (AOR: 425.4; 95 % CI: 9.6–18,808), younger mothers’age (AOR: 49.55;95 % CI: 1.59–1544), parents’negative attitude towards immunization (AOR: 20.2; 95 % CI: 1.46–278.9), and poorerparents’exposure to information on vaccination (AOR: 28.07; 95 % CI: 2.26–348.1). Longer distance from the vaccinationcenters was marginally significant (p=0.05).Conclusion:Vaccination coverage was high; however, 1 out of 7 children was partially vaccinated, and 1 out of 4 didnot complete timely the recommended vaccinations. In order to improve the immunization coverage, it is necessary tostrengthen ANC services, and to improve parents’information and attitude towards immunization, targeting youngerparents and families living far away from vaccination centers, using appropriate communication strategies. Finally, theestimated OPV-3 coverage is reassuring in relation to the ongoing polio outbrea