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

Subsets of groups in public-key cryptography

We suggest the usage of algebraic subsets instead of subgroups in public-key cryptography. In particular, we present the subset version of two protocols introduced by Shpilrain and Ushakov with some examples in ascending HNN-extensions of free-abelian groups and discuss their resistance to length and distance based attacks. We also introduce several new group theoretic problems arising from this work.Comment: 14 pages, comments welcome

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