1,200 research outputs found
Lattice congruences, fans and Hopf algebras
We give a unified explanation of the geometric and algebraic properties of
two well-known maps, one from permutations to triangulations, and another from
permutations to subsets. Furthermore we give a broad generalization of the
maps. Specifically, for any lattice congruence of the weak order on a Coxeter
group we construct a complete fan of convex cones with strong properties
relative to the corresponding lattice quotient of the weak order. We show that
if a family of lattice congruences on the symmetric groups satisfies certain
compatibility conditions then the family defines a sub Hopf algebra of the
Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has
a basis which is described by a type of pattern-avoidance. Applying these
results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite
sequence of smaller algebras, where the second algebra in the sequence is the
Hopf algebra of non-commutative symmetric functions. We also associate both a
fan and a Hopf algebra to a set of permutations which appears to be
equinumerous with the Baxter permutations.Comment: 34 pages, 1 figur
A characterization of covering equivalence
Let A={a_s(mod n_s)}_{s=1}^k and B={b_t(mod m_t)}_{t=1}^l be two systems of
residue classes. If |{1\le s\le k: x=a_s (mod n_s)}| and |{1\le t\le l: x=b_t
(mod m_t)}| are equal for all integers x, then A and B are said to be covering
equivalent. In this paper we characterize the covering equivalence in a simple
and new way. Using the characterization we partially confirm a conjecture of R.
L. Graham and K. O'Bryant
Lattice congruences of the weak order
We study the congruence lattice of the poset of regions of a hyperplane
arrangement, with particular emphasis on the weak order on a finite Coxeter
group. Our starting point is a theorem from a previous paper which gives a
geometric description of the poset of join-irreducibles of the congruence
lattice of the poset of regions in terms of certain polyhedral decompositions
of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let
\eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show
that the fibers of \eta_K constitute the smallest lattice congruence with
1\equiv s for every s\in(S-K). We give an algorithm for determining the
congruence lattice of the weak order for any finite Coxeter group and for a
finite Coxeter group of type A or B we define a directed graph on subsets or
signed subsets such that the transitive closure of the directed graph is the
poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure
Cambrian Lattices
For an arbitrary finite Coxeter group W we define the family of Cambrian
lattices for W as quotients of the weak order on W with respect to certain
lattice congruences. We associate to each Cambrian lattice a complete fan,
which we conjecture is the normal fan of a polytope combinatorially isomorphic
to the generalized associahedron for W. In types A and B we obtain, by means of
a fiber-polytope construction, combinatorial realizations of the Cambrian
lattices in terms of triangulations and in terms of permutations. Using this
combinatorial information, we prove in types A and B that the Cambrian fans are
combinatorially isomorphic to the normal fans of the generalized associahedra
and that one of the Cambrian fans is linearly isomorphic to Fomin and
Zelevinsky's construction of the normal fan as a "cluster fan." Our
construction does not require a crystallographic Coxeter group and therefore
suggests a definition, at least on the level of cellular spheres, of a
generalized associahedron for any finite Coxeter group. The Tamari lattice is
one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are
identified and characterized in terms of signed pattern avoidance. We also show
that open intervals in Cambrian lattices are either contractible or homotopy
equivalent to spheres.Comment: Revisions in exposition (partly in response to the suggestions of an
anonymous referee) including many new figures. Also, Conjecture 1.4 and
Theorem 1.5 are replaced by slightly more detailed statements. To appear in
Adv. Math. 37 pages, 8 figure
Intermediate problems in modular circuits satisfiability
In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite
algebras had been introduced and applied to sketch P versus NP-complete
borderline for circuits satisfiability over algebras from congruence modular
varieties. However the problem for nilpotent (which had not been shown to be
NP-hard) but not supernilpotent algebras (which had been shown to be polynomial
time) remained open.
In this paper we provide a broad class of examples, lying in this grey area,
and show that, under the Exponential Time Hypothesis and Strong Exponential
Size Hypothesis (saying that Boolean circuits need exponentially many modular
counting gates to produce boolean conjunctions of any arity), satisfiability
over these algebras have intermediate complexity between and , where measures how much a nilpotent algebra
fails to be supernilpotent. We also sketch how these examples could be used as
paradigms to fill the nilpotent versus supernilpotent gap in general.
Our examples are striking in view of the natural strong connections between
circuits satisfiability and Constraint Satisfaction Problem for which the
dichotomy had been shown by Bulatov and Zhuk
Satisfiability in multi-valued circuits
Satisfiability of Boolean circuits is among the most known and important
problems in theoretical computer science. This problem is NP-complete in
general but becomes polynomial time when restricted either to monotone gates or
linear gates. We go outside Boolean realm and consider circuits built of any
fixed set of gates on an arbitrary large finite domain. From the complexity
point of view this is strictly connected with the problems of solving equations
(or systems of equations) over finite algebras.
The research reported in this work was motivated by a desire to know for
which finite algebras there is a polynomial time algorithm that
decides if an equation over has a solution. We are also looking for
polynomial time algorithms that decide if two circuits over a finite algebra
compute the same function. Although we have not managed to solve these problems
in the most general setting we have obtained such a characterization for a very
broad class of algebras from congruence modular varieties. This class includes
most known and well-studied algebras such as groups, rings, modules (and their
generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie
algebras), lattices (and their extensions like Boolean algebras, Heyting
algebras or other algebras connected with multi-valued logics including
MV-algebras).
This paper seems to be the first systematic study of the computational
complexity of satisfiability of non-Boolean circuits and solving equations over
finite algebras. The characterization results provided by the paper is given in
terms of nice structural properties of algebras for which the problems are
solvable in polynomial time.Comment: 50 page
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