134 research outputs found
On the power pseudovariety
Some new semantic and syntactic characterizations of the members of the power
pseudovariety are obtained. This leads in particular to new
algorithms for deciding membership in
Solution to the Henckell--Rhodes problem: finite -inverse covers do exist
For a finite connected graph with set of edges , a finite
-generated group is constructed such that the set of relations
satisfied by (with a word over ) is closed under deletion
of generators (i.e.~edges). As a consequence, every element admits a
unique minimal set of edges (the \emph{content} of ) needed
to represent as a word over . The
crucial property of the group is that connectivity in the graph
is encoded in in the following sense: if a word forms a
path in then there exists a -equivalent
word which also forms a path and uses only edges from
their content; in particular, the content of the corresponding group element
spans a connected subgraph of containing the
vertices and . As an application it is shown that every finite inverse
monoid admits a finite -inverse cover. This solves a long-standing problem
of Henckell and Rhodes.Comment: 46 pages, 14 figures, new result Cor. 2.7 included, several
inaccuracies removed, more details include
The algebra of adjacency patterns: Rees matrix semigroups with reversion
We establish a surprisingly close relationship between universal Horn classes
of directed graphs and varieties generated by so-called adjacency semigroups
which are Rees matrix semigroups over the trivial group with the unary
operation of reversion. In particular, the lattice of subvarieties of the
variety generated by adjacency semigroups that are regular unary semigroups is
essentially the same as the lattice of universal Horn classes of reflexive
directed graphs. A number of examples follow, including a limit variety of
regular unary semigroups and finite unary semigroups with NP-hard variety
membership problems.Comment: 30 pages, 9 figure
Closures of regular languages for profinite topologies
The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the
pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic omega-semigroup. It is inherited by a
subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety A, as well as some of
its subpseudovarieties are shown to be full. The interest in such descriptions stems from the fact that, for each of the main pseudovarieties V in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in V. In the cases of A and of the pseudovariety DA of semigroups in which all regular elements are idempotents, this is a new result.PESSOA French-Portuguese project
Egide-Grices 11113YM, "Automata, profinite semigroups and symbolic dynamics".FCT -- Fundação para a Ciência e a Tecnologia, respectively under the projects
PEst-C/MAT/UI0144/2011 and PEst-C/MAT/UI0013/2011.ANR 2010 BLAN 0202 01 FREC.AutoMathA programme of the European Science Foundation.FCT and the project PTDC/MAT/65481/2006 which was partly funded by the European Community Fund FEDER
The finite basis problem for Kauffman monoids
We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid Kn are nonfinitely based for each n≥3. This result holds also for the case when Kn is considered as an involution semigroup under either of its natural involutions. © 2015, Springer Basel
Reducibility of joins involving some locally trivial pseudovarieties
In this paper, we show that sigma-reducibility is preserved under joins with K, where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with D, the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that sigma-reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product, we prove that Sl*K is k-tame, where Sl stands for the pseudovariety of semilattices.FCT through the Centro de Matemática da Universidade do MinhoEuropean Community Fund FEDE
Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory
We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research
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