Presentations for monoids of finite partial isometries

In this paper we give presentations for the monoid $\mathcal{DP}_n$ of all partial isometries on $\{1,\ldots,n\}$ and for its submonoid $\mathcal{ODP}_n$ of all order-preserving partial isometries.Comment: 11 pages, submitte

A new algebraic invariant for weak equivalence of sofic subshifts

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. Two independent approaches are used: ζ-semigroups as recognition structures for sofic subshifts, and relatively free profinite semigroups. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts and aperiodic subshifts. The algebraic invariant is compared with other robust conjugacy invariants.Research programme AutoMathA of ESF; Pessoa bilateral project Egide/Grices 11113YM "Automata, profinite semigroups and symbolic dynamics"; FCT, grant SFRH/BD/24200/2005; POCI 2010; FS

Normally ordered semigroups

Glasgow Mathematical Journal, nº 50 (2008), p. 325-333In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered blockgroups forms a pseudovariety of semigroups and, by using theMunn representation of a block-group, we deduce the decompositions in Mal’cev products NOS = EI m POI and NOS \ A = N m POI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing that BG = EI m Ecom = N m Ecom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively

On the power pseudovariety PCS\mathbf{PCS}

Some new semantic and syntactic characterizations of the members of the power pseudovariety $\mathbf{PCS}$ are obtained. This leads in particular to new algorithms for deciding membership in $\mathbf{PCS}$