819 research outputs found
On the logical definability of certain graph and poset languages
We show that it is equivalent, for certain sets of finite graphs, to be
definable in CMS (counting monadic second-order logic, a natural extension of
monadic second-order logic), and to be recognizable in an algebraic framework
induced by the notion of modular decomposition of a finite graph. More
precisely, we consider the set of composition operations on graphs
which occur in the modular decomposition of finite graphs. If is a subset
of , we say that a graph is an \calF-graph if it can be
decomposed using only operations in . A set of -graphs is recognizable if
it is a union of classes in a finite-index equivalence relation which is
preserved by the operations in . We show that if is finite and its
elements enjoy only a limited amount of commutativity -- a property which we
call weak rigidity, then recognizability is equivalent to CMS-definability.
This requirement is weak enough to be satisfied whenever all -graphs are
posets, that is, transitive dags. In particular, our result generalizes Kuske's
recent result on series-parallel poset languages
Universal enveloping algebras and universal derivations of Poisson algebras
Let be an arbitrary field of characteristic . It is shown that for any
the universal enveloping algebras of the Poisson symplectic algebra
and the Weyl algebra are isomorphic and the canonical
isomorphism between them easily leads to the Moyal product.
A basis of the universal enveloping algebra of a free Poisson algebra
is constructed and proved that the left dependency of a
finite number of elements of over is algorithmically recognizable.
We prove that if two elements of a free Poisson algebra do not generate a free
two generated subalgebra then they commute. The Fox derivatives on free Poisson
algebras are defined and it is proved that an analogue of the Jacobian
Conjecture for two generated free Poisson algebras is equivalent to the
two-dimensional classical Jacobian Conjecture. A new proof of the tameness of
automorphisms of two generated free Poisson algebras is also given.Comment: 20 page
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