Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so provide powerful combinatorial tools to work with them. This paper constructs an analogous `quasi-crystal' structure for the hypoplactic monoid, whose elements can be identified with quasi-ribbon tableaux and whose connection with the theory of quasi-symmetric functions echoes the connection of the plactic monoid with the theory of symmetric functions. This quasi-crystal structure and the associated quasi-Kashiwara operators are shown to interact just as neatly with the combinatorics of quasi-ribbon tableaux and with the hypoplactic version of the Robinson--Schensted--Knuth correspondence. A study is then made of the interaction of the crystal graph of the plactic monoid and the quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal structure is applied to prove some new results about the hypoplactic monoid.Comment: 55 pages. Minor revision to fix typos, add references, and discuss an open questio

Analysis of engaged online collaborative discourse: a methodological approach

The purpose of this chapter is to present a reflection on collaborative learning mediated by the computer, discussing some difficulties and methodological constraints that we encounter when we try to analyze the interactions that occurred in this collaboration in an online course and the level of involvement in ollaborative discourse produced by participants. For we apply the Speech Involvement Scale Collaborative Computer-mediated Conference.Projeto MEDEIAinfo:eu-repo/semantics/publishedVersio

Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, $p$-conjugacy is "almost" transitive, $\sim_c$ is strictly included in $\sim_p$, and the $p$- and $c$-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid $M$ defined by a finite complete presentation such that the $c$-conjugacy problem for $M$ is undecidable, and that for finitely presented monoids, the $c$-conjugacy problem and the word problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091

On finite complete presentations and exact decompositions of semigroups

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation. It is also proved that a semigroup M 0[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation

Idempotent Varieties of Incidence Monoids and Bipartite Posets

090ENH-21The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green’s relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same -class and their multiplication by an idempotent of the same -class gives conjugate elements in the group.epub_ahead_of_prin

Incidente tático-policial versus atirador ativo

Atualmente, a Polícia de Segurança Pública trata a problemática dos ITP’s e ITP’s com atiradores ativos num único documento, a NEP n.º DN/AUOOS/DO/01/11, de 05/01/2009, e o seu Anexo E, de 13/12/2012. Face às particularidades existentes nos dois cenários, este único documento que se pretende esclarecedor para o patrulheiro, apresenta duas formas distintas de atuação. É neste puzzle que se encontram os fundamentos para a redação da presente investigação. Considerando as inegáveis diferenças existentes entre ambas as realidades, que obrigam à definição de diferentes formas de resolução, seria de se esperar que também no campo concetual existisse uma separação ao nível do enquadramento. Assim, desenvolveu-se uma análise teórica de ambos os conceitos, sustentada, por um lado, na realidade vivida nos Estados Unidos da América e toda a produção literária relacionada com o tema dos atiradores ativos aí produzida e, por outro lado, no suporte doutrinário sobre a problemática dos incidentes tático-policiais e toda a regulamentação existente ao nível doméstico. Em segundo plano, procurou-se recolher argumentos junto de especialistas, através de um estudo exploratório baseado em entrevistas abertas, que nos ajudassem a colmatar algumas das lacunas existentes. Depois do levantamento do “estado da arte” e da recolha de novos elementos de fontes primárias, foi possível sugerir uma definição própria para atirador ativo. Tais elementos permitiram ainda propor uma recomendação de melhoria na forma de enquadramento destes assuntos, porquanto, os dados apontam para a existência de uma divergência a nível concetual.Currently, the Polícia de Segurança Pública deals with the tactical incidents and active shooter´s subject, in the same internal document: the NEP n.º DN/AUOOS/DO/01/11, dated 05/01/2009, and its Annex E, dated 13/12/2012. Considering the particularities of those both scenarios, this document should be as clear as possible for the police officer on the field, as it shows us two different ways of action. The elements to write this research are based on this puzzle, specifically considering all the differences that exists, that forces us against two different ways of resolution, it would be expected to have those concepts in separate doctrinaire supports. Therefore, we developed a theoretical analysis of both subjects, supported, on one hand, by the reality of the United States of America and all its literary production related with this subject and, on the other hand, based on the doctrinaire that exists in our country related with the tactical incidents. Behind the stage, we try to assemble arguments to solve the incipiency of doctrine by interviewing specialists on the subject. After gathering all the elements of the “state of the art” and the collection of new fundamentals of primary sources, it was possible to suggest an active shooter definition by our own. Those elements allowed us to suggest a recommendation to improve the way we look at this two subjects, as the majority of the data collected points towards two different conceptual approaches.N/