3,835 research outputs found
The Structure of a Graph Inverse Semigroup
Given any directed graph E one can construct a graph inverse semigroup G(E),
where, roughly speaking, elements correspond to paths in the graph. In this
paper we study the semigroup-theoretic structure of G(E). Specifically, we
describe the non-Rees congruences on G(E), show that the quotient of G(E) by
any Rees congruence is another graph inverse semigroup, and classify the G(E)
that have only Rees congruences. We also find the minimum possible degree of a
faithful representation by partial transformations of any countable G(E), and
we show that a homomorphism of directed graphs can be extended to a
homomorphism (that preserves zero) of the corresponding graph inverse
semigroups if and only if it is injective.Comment: 19 pages; corrected errors, improved organization, strengthened a
result (Theorem 20), added reference
Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations
A partial algebra construction of Gr\"atzer and Schmidt from
"Characterizations of congruence lattices of abstract algebras" (Acta Sci.
Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to
a well-known fact that every finite distributive lattice is representable, seen
as a special case of the Finite Lattice Representation Problem.
The construction of this proof brings together Birkhoff's representation
theorem for finite distributive lattices, an emphasis on boolean lattices when
representing finite lattices, and a perspective based on inequalities of
partially ordered sets. It may be possible to generalize the techniques used in
this approach.
Other than the aforementioned representation theorem only elementary tools
are used for the two theorems of this note. In particular there is no reliance
on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and
Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice
to be representable (see William J. DeMeo), such as the closure method
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
Congruences for Taylor expansions of quantum modular forms
Recently, a beautiful paper of Andrews and Sellers has established linear
congruences for the Fishburn numbers modulo an infinite set of primes. Since
then, a number of authors have proven refined results, for example, extending
all of these congruences to arbitrary powers of the primes involved. Here, we
take a different perspective and explain the general theory of such congruences
in the context of an important class of quantum modular forms. As one example,
we obtain an infinite series of combinatorial sequences connected to the
"half-derivatives" of the Andrews-Gordon functions and with Kashaev's invariant
on torus knots, and we prove conditions under which the sequences
satisfy linear congruences modulo at least of primes of primes
Non-existence of Ramanujan congruences in modular forms of level four
Ramanujan famously found congruences for the partition function like p(5n+4)
= 0 modulo 5. We provide a method to find all simple congruences of this type
in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is
non-vanishing on the upper half plane. This is applied to answer open questions
about the (non)-existence of congruences in the generating functions for
overpartitions, crank differences, and 2-colored F-partitions.Comment: 19 page
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