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Automaton semigroups: New constructions results and examples of non-automaton semigroups
Publication . Brough, Tara; Cain, Alan J.; CMA - Centro de Matemática e Aplicações; Elsevier
This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise as automaton semigroups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. We also consider closure under certain kinds of Rees matrix constructions, strong semilattices, and small extensions. Finally, we prove that no subsemigroup of (N,+) arises as an automaton semigroup. (Previously, (N,+) itself was the unique example of a semigroup having the ‘general’ properties of automaton semigroups (such as residual finiteness, solvable word problem, etc.) but that was known not to arise as an automaton semigroup.)
Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms
Publication . Alves, Carlos J. S.; Martins, Nuno F. M.; Valtchev, Svilen S.; DM - Departamento de Matemática; CMA - Centro de Matemática e Aplicações; Elsevier Science B.V., Amsterdam.
The direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains.
Conjugacy in inverse semigroups
Publication . Araújo, João; Kinyon, Michael; Konieczny, Janusz; DM - Departamento de Matemática; CMA - Centro de Matemática e Aplicações; Elsevier
In a group G, elements a and b are conjugate if there exists g∈G such that g−1ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼ib, if there exists g∈S1 such that g−1ag=b and gbg−1=a. The purpose of this paper is to study the conjugacy ∼i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.
Context-free word problem semigroups
Publication . Brough, Tara; Cain, Alan J.; Pfeiffer, Markus; CMA - Centro de Matemática e Aplicações
This paper studies the classes of semigoups and monoids with context-free and deterministic context-free word problem. First, some examples are exhibited to clarify the relationship between these classes and their connection with the notions of word-hyperbolicity and automaticity. Second, a study is made of whether these classes are closed under applying certain semigroup constructions, including direct products and free products, or under regressing from the results of such constructions to the original semigroup(s) or monoid(s).
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Fundação para a Ciência e a Tecnologia
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5876
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UID/Multi/04621/2013