54 research outputs found
Representations and identities of plactic-like monoids
Funding Information: The first and fourth authors were supported by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Centre for Mathematics and Applications) and PTDC/MAT-PUR/31174/2017. Publisher Copyright: © 2022We exhibit faithful representations of the hypoplactic, stalactic, taiga, sylvester, Baxter and right patience sorting monoids of each finite rank as monoids of upper triangular matrices over any semiring from a large class including the tropical semiring and fields of characteristic 0. By analysing the image of these representations, we show that the variety generated by a single hypoplactic (respectively, stalactic or taiga) monoid of rank at least 2 coincides with the variety generated by the natural numbers together with a fixed finite monoid H (respectively, F) and forms a proper subvariety of the variety generated by the plactic monoid of rank 2.publishersversionpublishe
Subword complexes and nil-Hecke moves
For a finite Coxeter group W, a subword complex is a simplicial complex
associated with a pair (Q, \rho), where Q is a word in the alphabet of simple
reflections, \rho is a group element. We describe the transformations of such a
complex induced by nil-moves and inverse operations on Q in the nil-Hecke
monoid corresponding to W. If the complex is polytopal, we also describe such
transformations for the dual polytope. For W simply-laced, these descriptions
and results of \cite{Go} provide an algorithm for the construction of the
subword complex corresponding to (Q, \rho) from the one corresponding to
(\delta(Q), \rho), for any sequence of elementary moves reducing the word Q to
its Demazure product \delta(Q). The former complex is spherical if and only if
the latter one is the (-1)-sphere.Comment: 6 pages. Comments welcome! arXiv admin note: substantial text overlap
with arXiv:1305.5499; and text overlap with arXiv:1111.3349 by other author
Tropical Representations and Identities of the Stylic Monoid
We exhibit a faithful representation of the stylic monoid of every finite
rank as a monoid of upper unitriangular matrices over the tropical semiring.
Thus, we show that the stylic monoid of finite rank generates the
pseudovariety , which corresponds to the class of
all piecewise testable languages of height , in the framework of Eilenberg's
correspondence. From this, we obtain the equational theory of the stylic
monoids of finite rank, show that they are finitely based if and only if , and that their identity checking problem is decidable in linearithmic
time. We also establish connections between the stylic monoids and other
plactic-like monoids, and solve the finite basis problem for the stylic monoid
with involution.Comment: 22 pages. Added results on the finite basis problem for the stylic
monoid with involution and updated reference
Identities and bases in plactic, hypoplactic, sylvester, and related monoids
The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep
connections to several areas of mathematics, in particular, to the theory of symmetric
functions. An active research topic is the identities satisfied by the plactic monoids of
finite rank. It is known that there is no “global" identity satisfied by the plactic monoid
of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic
monoid (the monoid of quasi-ribbon tableaux), sylvester monoid (the monoid of binary
search trees) and Baxter monoid (the monoid of pairs of twin binary search trees), satisfy
global identities, and the shortest identities have been characterized.
In this thesis, we present new results on the identities satisfied by the hypoplactic,
sylvester, #-sylvester and Baxter monoids. We show how to embed these monoids, of any
rank strictly greater than 2, into a direct product of copies of the corresponding monoid
of rank 2. This confirms that all monoids of the same family, of rank greater than or equal
to 2, satisfy exactly the same identities. We then give a complete characterization of those
identities, thus showing that the identity checking problems of these monoids are in the
complexity class P, and prove that the varieties generated by these monoids have finite
axiomatic rank, by giving a finite basis for them. We also give a subdirect representation
ofmultihomogeneous monoids by finite subdirectly irreducible Rees factor monoids, thus
showing that they are residually finite.O ubíquo monóide plático, também conhecido como o monóide dos diagramas de Young,
tem ligações profundas a várias áreas de Matemática, em particular à teoria das funções
simétricas. Um tópico de pesquisa ativo é o das identidades satisfeitas pelos monóides
pláticos de característica finita. Sabe-se que não existe nenhuma identidade “global” satisfeita
pelos monóides pláticos de cada característica. Em contraste, sabe-se que monóides
ligados ao monóide plático, como o monóide hipoplático (o monóide dos diagramas quasifita),
o monóide silvestre (o monóide de árvores de busca binárias) e o monóide de Baxter
(o monóide de pares de árvores de busca binária gémeas), satisfazem identidades globais,
e as identidades mais curtas já foram caracterizadas.
Nesta tese, apresentamos novos resultados acerca das identidades satisfeitas pelos monóides
hipopláticos, silvestres, silvestres-# e de Baxter. Mostramos como mergulhar estes
monóides, de característica estritamente maior que 2, num produto direto de cópias do
monóide correspondente de característica 2. Confirmamos assim que todos os monóides
da mesma família, de característica maior ou igual a 2, satisfazem exatamente as mesmas
identidades. A seguir, damos uma caracterização completa dessas identidades, mostrando
assim que os problemas de verificação de identidades destes monóides estão na classe de
complexidade P, e provamos que as variedades geradas por estes monóides têm característica
axiomática finita, ao apresentar uma base finita para elas. Também damos uma
representação subdireta de monóides multihomogéneos por monóides fatores de Rees
finitos e subdiretamente irredutíveis, mostrando assim que são residualmente finitos
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