180 research outputs found
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
A Groupoid Approach to Discrete Inverse Semigroup Algebras
Let be a commutative ring with unit and an inverse semigroup. We show
that the semigroup algebra can be described as a convolution algebra of
functions on the universal \'etale groupoid associated to by Paterson. This
result is a simultaneous generalization of the author's earlier work on finite
inverse semigroups and Paterson's theorem for the universal -algebra. It
provides a convenient topological framework for understanding the structure of
, including the center and when it has a unit. In this theory, the role of
Gelfand duality is replaced by Stone duality.
Using this approach we are able to construct the finite dimensional
irreducible representations of an inverse semigroup over an arbitrary field as
induced representations from associated groups, generalizing the well-studied
case of an inverse semigroup with finitely many idempotents. More generally, we
describe the irreducible representations of an inverse semigroup that can
be induced from associated groups as precisely those satisfying a certain
"finiteness condition". This "finiteness condition" is satisfied, for instance,
by all representations of an inverse semigroup whose image contains a primitive
idempotent
M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities
We generalize the character formulas for multiplicities of irreducible
constituents from group theory to semigroup theory using Rota's theory of
M\"obius inversion. The technique works for a large class of semigroups
including: inverse semigroups, semigroups with commuting idempotents,
idempotent semigroups and semigroups with basic algebras. Using these tools we
are able to give a complete description of the spectra of random walks on
finite semigroups admitting a faithful representation by upper triangular
matrices over the complex numbers. These include the random walks on chambers
of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and
Diaconis. Applications are also given to decomposing tensor powers and exterior
products of rook matrix representations of inverse semigroups, generalizing and
simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update
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
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