64 research outputs found
Radical theory for group semiautomata
A Kurosh-Amitsur radical theory is developed for group semiautomata. Radical theory stems from ring theory, it is apt for deriving structure theorems and for a comparative study of properties. Unlikely to conventional radical theories, the radical of a group semiautomaton need not be a subsemiautomaton, so the whole scene will take place in a suitably constructed category. The fundamental facts of the theory are described in § 2. A special feature of the theory, the existence of complementary radicals, is discussed in § 3. Restricting the theory to additive automata, which still comprise linear sequential machines, in § 4 stronger results will be achieved, and also a (sub)direct decomposition theorem for certain semisimple group semiautomata will be proved. Examples are given at appropriate places. The paper may serve also as a framework for future structural investigations of group semiautomata
Complex Algebras of Arithmetic
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a
sequence of arithmetic and logical operations to be performed on sets of
natural numbers. Arithmetic circuits can also be viewed as the elements of the
smallest subalgebra of the complex algebra of the semiring of natural numbers.
In the present paper, we investigate the algebraic structure of complex
algebras of natural numbers, and make some observations regarding the
complexity of various theories of such algebras
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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