35 research outputs found
A minimal nonfinitely based semigroup whose variety is polynomially recognizable
We exhibit a 6-element semigroup that has no finite identity basis but
nevertheless generates a variety whose finite membership problem admits a
polynomial algorithm.Comment: 16 pages, 3 figure
The Finite Basis Problem for Kiselman Monoids
In an earlier paper, the second-named author has described the identities
holding in the so-called Catalan monoids. Here we extend this description to a
certain family of Hecke--Kiselman monoids including the Kiselman monoids
. As a consequence, we conclude that the identities of
are nonfinitely based for every and exhibit a finite
identity basis for the identities of each of the monoids and
.
In the third version a question left open in the initial submission has beed
answered.Comment: 16 pages, 1 table, 1 figur
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
Monoid varieties with extreme properties
Finite monoids that generate monoid varieties with uncountably many
subvarieties seem rare, and surprisingly, no finite monoid is known to generate
a monoid variety with countably infinitely many subvarieties. In the present
article, it is shown that there are, nevertheless, many finite monoids that
generate monoid varieties with continuum many subvarieties; these include any
finite inherently non-finitely based monoid and any monoid for which is
an isoterm. It follows that the join of two Cross monoid varieties can have a
continuum cardinality subvariety lattice that violates the ascending chain
condition.
Regarding monoid varieties with countably infinitely many subvarieties, the
first example of a finite monoid that generates such a variety is exhibited. A
complete description of the subvariety lattice of this variety is given. This
lattice has width three and contains only finitely based varieties, all except
two of which are Cross
A Minimal Nonfinitely Based Semigroup Whose Variety is Polynomially Recognizable
We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm. © 2011 Springer Science+Business Media, Inc.Acknowledgement. The first and the second authors acknowledge support from the Federal Education Agency of Russia, project 2.1.1/3537, and from the Russian Foundation for Basic Research, grant 09-01-12142