12,414 research outputs found
On the Lie enveloping algebra of a post-Lie algebra
We consider pairs of Lie algebras and , defined over a common
vector space, where the Lie brackets of and are related via a
post-Lie algebra structure. The latter can be extended to the Lie enveloping
algebra . This permits us to define another associative product on
, which gives rise to a Hopf algebra isomorphism between and
a new Hopf algebra assembled from with the new product.
For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari
order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths
or (m+1)-ary trees. On another hand, the Tamari order is related to the product
in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new
combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is
described by the m-Tamari lattices.
In the same way as planar binary trees can be interpreted as sylvester
classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what
we call m-permutations. These objects are no longer in bijection with
decreasing (m+1)-ary trees, and a finer congruence, called metasylvester,
allows us to build Hopf algebras based on these decreasing trees. At the
opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of
graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions
and quasi-symmetric functions in a natural way. Finally, the algebras of packed
words and parking functions also admit such m-analogues, and we present their
subalgebras and quotients induced by the various congruences.Comment: 51 page
Propositional logic with short-circuit evaluation: a non-commutative and a commutative variant
Short-circuit evaluation denotes the semantics of propositional connectives
in which the second argument is evaluated only if the first argument does not
suffice to determine the value of the expression. Short-circuit evaluation is
widely used in programming, with sequential conjunction and disjunction as
primitive connectives.
We study the question which logical laws axiomatize short-circuit evaluation
under the following assumptions: compound statements are evaluated from left to
right, each atom (propositional variable) evaluates to either true or false,
and atomic evaluations can cause a side effect. The answer to this question
depends on the kind of atomic side effects that can occur and leads to
different "short-circuit logics". The basic case is FSCL (free short-circuit
logic), which characterizes the setting in which each atomic evaluation can
cause a side effect. We recall some main results and then relate FSCL to MSCL
(memorizing short-circuit logic), where in the evaluation of a compound
statement, the first evaluation result of each atom is memorized. MSCL can be
seen as a sequential variant of propositional logic: atomic evaluations cannot
cause a side effect and the sequential connectives are not commutative. Then we
relate MSCL to SSCL (static short-circuit logic), the variant of propositional
logic that prescribes short-circuit evaluation with commutative sequential
connectives.
We present evaluation trees as an intuitive semantics for short-circuit
evaluation, and simple equational axiomatizations for the short-circuit logics
mentioned that use negation and the sequential connectives only.Comment: 34 pages, 6 tables. Considerable parts of the text below stem from
arXiv:1206.1936, arXiv:1010.3674, and arXiv:1707.05718. Together with
arXiv:1707.05718, this paper subsumes most of arXiv:1010.367
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