Assis Azevedo

On finitely based pseudovarieties of the forms V ∗ D and V ∗ Dn☆

Abstract Let D n be the pseudovariety of all finite semigroups in which products of length n are right zeros and let D = ⋃ n≥1 D n . It is shown in this paper that, if V is a pseudovariety of semigroups whose global g V is finitely based, then V ∗ D n ( n≥1 ) and V ∗ D are also finitely based. Moreover, if V is itself finitely based and contains the aperiodic five-element Brandt semigroup, then g V is also finitely based. As a further application, it is proved that the finite basis properties for g V , V ∗ D and V ∗ D n ( n≥1 ) are all equivalent for an arbitrary non-group monoidal pseudovariety V .

The join of the pseudovarieties of R-trivial and L-trivial monoids

Abstract The smallest pseudovariety containing all finite R -trivial or L -trivial monoids is described by means of a single effective pseudoidentity, thus establishing the decidability of its membership problem.

The N-membranes problem for quasilinear degenerate systems

We study the regularity of the solution of the variational inequality for the problem of N -membranes in equilibrium with a degenerate operator of p-Laplacian type, 1 < p < 1, for which we obtain the corresponding Lewy‐Stampacchia inequalities. By considering the problem as a system coupled through the characteristic functions of the sets where at least two membranes are in contact, we analyze the stability of the coincidence sets.

Implicit Operations on Certain Classes of Semigroups

Varieties of algebras are characterized by identities, where an identity is a formal equality of two terms (i.e., operations defined by means of the underlying operations). Analogously, pseudovarieties of (finite) algebras are defined by pseudo-identities, these being formal equalities of so-called implicit operations (briefly, functions compatible with all homomorphisms). To further explore this analogy to yield results on finite algebras, it is necessary to obtain clear descriptions of implicit operations. This work is a contribution to this project in the area of semigroup theory. All unary implicit operations on semigroups are described, and the implicit operations on certain pseudo-varieties of semigroups are given in terms of “generating” operations. The existence of some unusual implicit operations is established based on classical combinatorial theorems about words.

Globals of pseudovarieties of commutative semigroups : the finite basis problem, decidability and gaps

Whereas pseudovarieties of commutative semigroups are known to be nitely based, the globals of monoidal pseudovarieties of commutative semi-groups are shown to be nitely based (or of nite vertex rank) if and only if the index is 0, 1 or !. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains order isomorphic to the reals. 1. Introduction Building on ideas of J. Rhodes and others 15, 16], Tilson 17] introduced categories and semigroupoids (categories without local identities) as a tool for studying semidirect products of semigroups. Weil and the rst author 10] integrated into Tilsons theory the proonite perspective culminating in the description of a basis of

PSEUDOVARIETY JOINS INVOLVING

J. Rhodes asked during the Chico Conference in 1986 for the calculation of joins of semigroup pseudovarieties. This paper proves that the join J∨H of the pseudovariety J of -trivial semigroups and of any 2-strongly decidable pseudovariety V of of completely regular semigroups is decidable. This problem was proposed by the first author for V=G, the pseudovariety of finite groups.

{\mathscr J}

Using a theorem of Reiterman, which characterizes pseudovarieties as classes of finite semigroups satisfying a set of pseudoidentities, and a characterization of the implict operations on DS, we calculate some joins of the form J∨ V, where V is a permutative pseudovariety. As a consequence we obtain that, for these V, J ∨ V is decidable if and only if V ∩ CS is decidable.

-TRIVIAL SEMIGROUPS

We consider the problem of finding the equilibrium position of two or three membranes constrained not to pass through each other. For general linear second order elliptic operators with measurable coefficients we prove the Lewy-Stampacchia type inequalities and we establish sufficient conditions on the external forces to obtain the stability of the coincidence sets of the membranes, in analogy with the obstacle problem.

The Join of the Pseudovariety J with Permutative Pseudovarieties

We study existence of solution of stationary quasivariational inequalities with gradient constraint and nonhomogeneous boundary condition of Neumann or Dirichlet type. Through two different approaches, one making use of a fixed point theorem and the other using a process of regularization and penalization, we obtain different sufficient conditions for the existence of solution.

Remarks on the two and three membranes problem

This article answers three questions of J. Almeida. Using combinatorial, algebraic and topological methods, we compute joins involving the pseudovariety of nite groups, the pseudovariety of semigroups in which each idempotent is a right zero and the pseudovariety generated by monoids M such that each idempotent of Mnf1g is a left zero. The need to organize nite semigroups into a hierarchy comes from several algorithmic problems in connection with computer science. The lattice of semigroup pseudovarieties (classes of nite semigroups closed under nite direct product, subsemigroup and homomorphic image) became the object of special consideration after the publication of Eilenberg’s treatise [11]. Many problems from language theory found indeed an interesting formulation within this scope. At the moment, one of the challenges is to understand some operators acting on pseudovarieties. In this perspective, topological approaches providing signican t results were developed during the last decade by Almeida. The present paper takes advantage of these techniques to answer three