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93
Leavitt path algebras of arbitrary graphs
 Houston J. Math
"... Abstract. We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ringtheoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of rowfinite graphs. Specifically, we identify those g ..."
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Cited by 41 (13 self)
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Abstract. We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ringtheoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of rowfinite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and semiprime. In our final result, we show that all Leavitt path algebras have zero Jacobson radical.
Cohomological quotients and smashing localizations
 Amer. J. Math
"... Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the coh ..."
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Cited by 24 (5 self)
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Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic Ktheory and demonstrates the relevance of the telescope
The Realization of InputOutput Maps Using Bialgebras
, 1991
"... We use the theory of bialgebras to provide the algebraic background for state space realization theorems for inputoutput maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent resul ..."
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Cited by 12 (7 self)
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We use the theory of bialgebras to provide the algebraic background for state space realization theorems for inputoutput maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent results involving analysis related to families of trees. If H is a bialgebra, we say that p H # is di#erentially produced by the algebra R with the augmentation # if there is right Hmodule algebra structure on R and there exists f h).
The origins of combinatorics on words
, 2007
"... We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early ..."
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Cited by 11 (0 self)
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We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a byproduct of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this twosided interaction.
Conway's Problem and the commutation of languages
 Bulletin of EATCS
, 2001
"... We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The se ..."
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Cited by 10 (5 self)
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We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1
Division algebras and noncommensurable isospectral manifolds
 Duke Math. J
"... Abstract. A. Reid [R] showed that if Γ1 and Γ2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different ..."
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Cited by 10 (1 self)
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Abstract. A. Reid [R] showed that if Γ1 and Γ2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different: there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by S. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants. 1.
Smarandache Rings
 MAGNA An international journal
, 2002
"... w. b. vasantha kandasamy smarandache rings a 5 a 3 a 8 a 7 a 16 a 14 a 18 a 1 a 0 a 2 american research press 2002 a 6 a 9 a 10 a 13 a 17 a 11 a 4 a 15 a 12 ..."
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Cited by 9 (0 self)
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w. b. vasantha kandasamy smarandache rings a 5 a 3 a 8 a 7 a 16 a 14 a 18 a 1 a 0 a 2 american research press 2002 a 6 a 9 a 10 a 13 a 17 a 11 a 4 a 15 a 12
Finitely generated subnormal subgroups of GLn(D) are central
 Journal of Algebra
, 2000
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ISOMORPHISMS BETWEEN LEAVITT ALGEBRAS AND THEIR MATRIX RINGS
, 2008
"... Abstract. Let K be any field, let Ln denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let Md(Ln) denote the ring of d × d matrices over Ln. In our main result, we show that Md(Ln) ∼ = Ln if and only if d and n − 1 are coprime. We use this isomorphism to answer a question ..."
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Cited by 9 (6 self)
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Abstract. Let K be any field, let Ln denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let Md(Ln) denote the ring of d × d matrices over Ln. In our main result, we show that Md(Ln) ∼ = Ln if and only if d and n − 1 are coprime. We use this isomorphism to answer a question posed in [14] regarding isomorphisms between various C*algebras. Furthermore, our result demonstrates that data about the K0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple Kalgebras.