Unary semigroups with an associate subgroup

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.Fundação para a Ciência e a Tecnologia (FCT

Matemática discreta

Publicação pedagógicaO presente livro é um texto de apoio à unidade curricular Matemática Discreta e tem assim o objectivo de ser uma apresentação simples, mas cuidada, de conceitos e resultados básicos da Teoria de Grafos e da Teoria de Números

Homomorphisms and congruences of medial semigroups with an associate subgroup

Let S be the model of a semigroup with an associate subgroup whose identity is a medial idempotent constructed by Blyth and Martins considered as a unary semigroup. For another such semigroup T, we construct all unary homomorphisms of S into T in terms of their parameters. On S we construct all unary congruences again directly from its parameters. This construction leads to a characterization of congruences in terms of kernels and traces. We describe the K, T, L, U and V relations on the lattice of all unary congruences on S, again in terms of parameters of S.Fundação para a Ciência e a Tecnologia (FCT

Evaluating delinquency policy interventions in Portugal

Social program and policies, interventions, and services, should be required to undergo rigorous systematic evaluation to address the policy question of how society should invest in the treatment of juvenile offenders in the institutional care system. Public policy decisions regarding programs for youths should be grounded on research-based knowledge and experience of academia and practitioners, program participants. Despite developments in intervention science, the existing empirical literature is seriously underdeveloped with respect to the issue of delinquency interventions and policies in Portugal. In other words, there is little systematic knowledge on the effects of existing policies in the youth system. In this paper, we address the available research and evidence on juvenile justice institutional interventions. We seek to explore the extent to which these are 1) informed by previous studies and 2) subject to analysis during and following policy termination.CIEC – Research Centre on Child Studies, IE, UMinho (FCT R&D unit 317), Portugal;National Funds through the FCT (Foundation for Science and Technology) and co-financed by European Regional Development Funds (FEDER) through the Competitiveness and Internationalization Operational Program (POCI) with the reference POCI-01-0145-FEDER-007562info:eu-repo/semantics/publishedVersio

Mathematical modelling and simulation of the physical system of pulleys in mathematical learning

The standards of the CDIO model (Conceiving, Designing, Implementing, and Operating) include the existence of project development experiences that involve different curricular courses and that provide team activities from the 1st year of engineering degrees. In this sense, it was implemented a teaching experience in the engineering degree in Electrical Power Systems at School of Engineering of the Porto Polytechnic (ISEP) based on a multidisciplinary project that involved modulation, simulation and construction of a prototype based on the concept of pulleys. Here we present the phases of the task development, the objectives to be achieved with its realisation, the results obtained by the students and their perceptions about the realization of the Project-based learning (PBL). Here we present the phases of the task development, the objectives to be achieved with its realisation, the results obtained by the students and their perceptions about the realization of the Project-based learning (PBL) challenge

Mathematical modelling and simulation of the pendulum physical system in mathematical learning

In the CDIO model and EUR-ACE certification, project development experiences providing team activities are foreseen since the 1st year of engineering courses. In order to evaluate the contribution of PBL activities in the learning of mathematical content, a multidisciplinary project was carried out in an engineering degree at School of the Porto Polytechnic (ISEP), which involved the construction and simulation based on the physical model of the concept of pendulum. To understand the meaning of the PBL task in the learning of mathematical contents from the students' perspective, we adopted a qualitative approach, collecting data from reports written by the students and their public oral presentations. We present the stages of the task development, the results obtained by the students and their perceptions of the PBL accomplishment. The results indicate that students recognize the usefulness of carrying out projects in their academic training and to prepare professional future

Prospective primary school teachers' activities when dealing with mathematics modelling tasks

The current teaching of mathematics is guided by recommendations that suggest the implementation of various activities in order to raise the understanding of mathematical knowledge. This diversity is related to the characteristics of the tasks proposed in the learning contexts. Among all tasks, the modelling ones call for the application of activities through different representations. So, it is important that teacher training courses promote experiences involving prospective teachers with this type of task. Based on this assumption, we intend to identify the activities that prospective primary school teachers perform in solving modelling tasks, the difficulties experienced in these tasks and the value of the models they determine. From the analysis of the resolutions of two tasks, we find that the prospective teachers translate the information of the data available in tables through graphs and analytical expressions. Some discuss models that determine which best fits the data. In the activities carried out, difficulties arise in determining the proportionality constant that best translates the problem situation, discussing the reasonableness of the values generated by the model, and sketching the graph of the model that best fits the experimental data. As for the usefulness of the model they determine, few prospective teachers are predicting outcomes.Pengajaran matematika saat ini dipandu oleh suatu rekomendasi yang menyarankan implementasi berbagai kegiatan untuk meningkatkan kemampuan pemahaman matematis. Keberagaman ini terkait dengan karakteristik tugas yang diusulkan dalam konteks pembelajaran. Di antara semua tugas, soal pemodelan membutuhkan aktivitas pembelajaran melalui representasi yang berbeda. Jadi, penting bahwa pelatihan guru mendorong pengalaman yang melibatkan para calon guru dengan jenis tugas ini. Berdasarkan asumsi ini, kami bermaksud mengidentifikasi kegiatan yang dilakukan calon guru sekolah dasar dalam menyelesaikan tugas pemodelan, kesulitan yang dialami dalam tugas ini, dan model yang mereka tentukan. Dari analisis terhadap dua tugas yang diberikan, kami menemukan bahwa calon guru menerjemahkan informasi dari data yang tersedia dalam bentuk tabel melalui grafik dan ekspresi analitis. Beberapa mendiskusikan model yang menentukan mana yang paling cocok dengan data. Dalam kegiatan yang dilakukan, kesulitan muncul dalam menentukan konstanta proporsionalitas yang paling baik menerjemahkan situasi masalah, membahas kewajaran nilai yang dihasilkan oleh model, dan membuat sketsa grafik model yang paling sesuai dengan data eksperimen. Adapun kegunaan model yang mereka tentukan, hanya sedikit diantara para calon guru yang memprediksi hasil dari permodelan yang diberikan.This work is funded by CIEd – Research Centre on Education, Institute of Education, University of Minho, projects UIDB/01661/2020 and UIDP/01661/2020, through national funds of FCT/MCTES-PT

The notion of function held by basic education pre-service teachers

The current curricular guidelines for mathematics education in Portugal emphasize the relevance of working with different representations of functions to promote understanding. Given this relevance, we seek understanding about the notion of function held by 37 basic education pre-service teachers in their first year of a master's course. Data were collected through a task focusing on identifying functions in situations based on diferent representations. The content analysis technique was then adopted in the search for an understanding of the justifications given by the participants. The results achieved suggest it is easier for the pre-service teachers to identify examples that are not functions than exemples that are functions. There is also a tendency for greater accuracy in the identification of examples expressed by tables than by algebraic expressions. The justifications presented show a notion of function as a relation between values of two non-empty sets, but without guaranteeing that this relation is single-valued.This research was funded by CIEd - Research Centre on Education, project UID/CED/01661/2019, Institute of Education, University of Minho, through national funds of FCT/MCTES-PT

Associate inverse subsemigroups of regular semigroups

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x ∈ S, in relation to the natural partial order ≤S. We describe the structure of a regular semigroup with an associate inverse subsemigroup, satisfying two natural conditions. As a articular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element.Fundação para a Ciência e a Tecnologia (FCT

Some orthodox monoids with associate inverse subsemigroups

By an associate inverse subsemigroup of a regular semigroup $S$ we mean a subsemigroup $T$ of $S$ containing a least associate of each $x \in S$, in relation to the natural partial order $\leq_S$ in $S$. In this paper we investigate a class of orthodox monoids with an associate inverse subsemigroup and obtain a known description of uniquely unit regular orthodox semigroups as a corollary. Also, by considering a more general situation, we identify the homomorphic image of a kind of semidirect product of a band with identity by an inverse monoid, thus extending a known result for unit regular orthodox semigroups.Portuguese Foundation for Science and Technology (FCT) through the Research Centre CMA