Showing papers on "Idempotence published in 2007"
Posted Content•
TL;DR: In this article, the authors describe a special class of representations of an inverse semigroup S on Hilbert's space which they call tight, and these representations are supported on a subset of the spectrum of the idempotent semilattice of S, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way.
Abstract: We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term "tight". These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the "tight spectrum", which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum.
We then treat the case of certain inverse semigroups constructed from a semigroupoid, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.
170 citations
Posted Content•
TL;DR: In this paper, an introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones.
Abstract: An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices.
90 citations
09 Jul 2007
TL;DR: Algebraic conditions on constraint languages Γ are presented that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL and it is shown that if C SP( Γ) is not first-order definable then it is L-hard.
Abstract: We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not first-order definable then it is L-hard. Our proofs rely on tame congruence theory and on a fine-grain analysis of the complexity of reductions used in the algebraic study of CSPs. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ) lies in P or is NP-complete and they match the recent classification of [1] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ) when the associated algebra of Γ is the idempotent reduct of a preprimal algebra.
64 citations
TL;DR: In this article, it was shown that two natural classes of semiring-semimodule pairs, the complete and the bi-inductive semiringsemimmodule pairs both give rise to iteration semiring semimodules.
Abstract: Conway semiring-module pairs and iteration semiring-semimodule pairs were shown to provide an axiomatic basis to automata on ω -words in [Bloom, Esik: Iteration Theories, Springer, 1993]. In this paper, we show that two natural classes of semiring-semimodule pairs, the complete and the bi-inductive semiring-semimodule pairs both give rise to iteration semiring-semimodule pairs. Complete semiring-semimodule pairs are defined by infinite sums and products, while a bi-inductive semiring-semimodule pair is an ordered semiring-semimodule pair possessing enough least pre-fixed points and greatest post-fixed points to solve linear inequations. Moreover, we show that when V is idempotent, then a semiring-semimodule pair equipped with a star and an omega operation satisfies the Conway equations (iteration semiring-semimodule pair equations, respectively) if and only if the quemiring associated with (S,V) embeds in a Conway semiring (iteration semiring, respectively).
44 citations
TL;DR: In this article, it was proved that the variety of representable idempotent commutative residuated lattices is locally finite and the n-generated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each.
Abstract: It is proved that the variety of representable idempotent commutative residuated lattices is locally finite The n-generated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each A constructive characterization of the subdirectly irreducible algebras is provided, with some applications The main result implies that every finitely based extension of positive relevance logic containing the mingle and Godel-Dummett axioms has a solvable deducibility problem
35 citations
Journal Article•
TL;DR: In this article, it was shown that any multiplicative generalized derivation of R is additive, i.e. g(xy)=g(x)y+xd(y) for all x, y in R and some derivationd of R.
Abstract: The purpose of this note is to prove the following. Suppose R is a ring having an idempotent elemente (e
eq 0, e
eq 1) which satisfies some conditions. If g is any multiplicative generalized derivation of R, i.e. g(xy)=g(x)y+xd(y), for all x, y in R and some derivationd of R, then g is additive.
34 citations
Journal Article•
TL;DR: Weighted Muller tree automata are introduced, over totally commutative complete semirings, acting on infinite trees, and it is shown that their behaviours coincide with the semantics of weighted restricted MSO-sentenees and the semanticsof weighted incomplete universal MSOs, provided that the underlying semiring is idempotent.
Abstract: We introduce weighted Muller tree automata, over totally commutative complete semirings, acting on infinite trees. We show that their behaviours coincide with the semantics of weighted restricted MSO-sentenees and the semantics of weighted incomplete universal MSO-sentences, provided that the underlying semiring is idempotent.
30 citations
TL;DR: In this paper, the spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric, and the authors deduce as a corollary of their main results the cyclic analogue of Helly's theorem.
Abstract: Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.
29 citations
Posted Content•
TL;DR: In this article, it was shown that a finite semiring of order > 2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a dense subsemiring of a finite idempotent commutative monoid.
Abstract: Our main result states that a finite semiring of order >2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a `dense' subsemiring of the endomorphism semiring of a finite idempotent commutative monoid.
We also investigate those subsemirings further, addressing e.g. the question of isomorphy.
28 citations
TL;DR: In this paper, it was shown that when the algebra has dimension greater than or equal to three, the idempotent rank equals the rank of the ideal of the monoid.
Abstract: In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.
27 citations
TL;DR: In this article, a stability theorem for the nullity of a linear combination c1P1 + c2P2 of two idempotent operators P1, P2 on a Banach space provided c 1, c 2 and c 1 + c 2 are nonzero is proved.
Abstract: We prove a stability theorem for the nullity of a linear combination c1P1 + c2P2 of two idempotent operators P1, P2 on a Banach space provided c1, c2 and c1 + c2 are nonzero. We then show that for c1P1 + c2P2 the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of c1, c2, and that the nullity, defect and index of c1P1 + c2P2 are stable.
TL;DR: In this article, the concrete form of every unital surjective map φ on a complex Banach space X such that AB is a non-zero idempotent if and only if φ ( A ) φ( B ) is for all A, B ∈ B ( X ) when the dimension of X is at least 3.
01 Jun 2007
TL;DR: In this paper, the authors give necessary and sufficient conditions for Sn−1 to be generated by idempotents over Euclidean domains and free left T -sets of finite rank, where T is a cancellative monoid in which every finitely generated left ideal is principal.
Abstract: If A is a stable basis algebra of rank n, then the set Sn−1 of endomorphisms of rank at most n − 1 is a subsemigroup of the endomorphism monoid of A. This paper gives a number of necessary and sufficient conditions for Sn−1 to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left T -sets of finite rank, where T is cancellative monoid in which every finitely generated left ideal is principal.
TL;DR: In this paper, the set of n × n idempotent matrices and linear operators that preserve these matrices over Boolean algebras were characterized. But the linear operators were not characterized for nonnegative integers and nonnegative reals.
Abstract: We consider the set of n × n idempotent matrices and we characterize the linear operators that preserve idempotent matrices over Boolean algebras. We also obtain characterizations of linear operators that preserve idempotent matrices over a chain semiring, the nonnegative integers and the nonnegative reals.
TL;DR: The first n - 1 projections forming the Jones tower of a subfactor generate a semisimple quotient of the Temperley-Lieb Algebra as mentioned in this paper, which can be represented pictorially by planar diagrams on n strings in a box, and these diagrams can be classified according to the number of non-through strings or gaps they have.
Abstract: The first n - 1 projections forming the Jones tower of a
II1 subfactor generate a semisimple quotient,TLn.., of the Temperley-
Lieb Algebra. This algebra can be represented pictorially by planar diagrams
on n strings in a box, and these diagrams can be classified according
to the number of non-through strings, or �gaps� they have. The
Jones-Wenzl Idempotent is the complement in TLn.. of the supremum
of the projections generating the Jones tower. We prove Ocneanu�s
formula for the coefficients of the one- and two-gap boxes in an explicit
expression of this element.
15 Aug 2007
TL;DR: It is shown how idempotence can be used to improve not only approximate, but also exact halfspace range searching, because these data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.
Abstract: Range searching is a well known problem in the area of geometric data structures. We consider this problem in the context of approximation, where an approximation parameter e > 0 is provided. Most prior work on this problem has focused on the case of relative errors, where each range shape R is bounded, and points within distance e ċ diam (R) of the range's boundary may or may not be included. We consider a different approximation model, called the absolute model, in which points within distance e of the range's boundary may or may not be included, regardless of the diameter of the range. We consider range spaces consisting of halfspaces, Euclidean balls, simplices, axis-aligned rectangles, and general convex bodies. We consider a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, and range emptiness. We show how idempotence can be used to improve not only approximate, but also exact halfspace range searching. Our data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.
01 Jan 2007
TL;DR: Convexity and cone-vexing abstractions Semen S. Kutateladze, G.L. Litvinov and G.B. Shpiz.
Abstract: convexity and cone-vexing abstractions Semen S. Kutateladze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Interval analysis for algorithms of idempotent and tropical mathematics Grigory L. Litvinov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Dequantization procedures related to the Maslov dequantization G.L. Litvinov and G.B. Shpiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
TL;DR: In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols.
Abstract: Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.
TL;DR: In this article, the authors prove an eigenvector existence theorem for linear operators on abstract idempotent spaces in the framework of the algebraic approach, which was known only for operators in free finite-dimensional semimodules.
Abstract: In this paper, we prove an eigenvector existence theorem for linear operators on abstract idempotent spaces in the framework of the algebraic approach. Earlier, an algebraic version of a similar statement was known only for operators in free finite-dimensional semimodules. The corresponding result for compact operators in semimodules of real continuous functions is known in the case of topological semimodules.
TL;DR: In this article, it was shown that under certain conditions, the idempotent elements of a hyperring form a hyperlattice and the orthogonal Idempotnet elements form a quasi-distributive hyperboolean algebra.
Abstract: In this paper we prove that under certain conditions, the idempotent elements of a hyperring form a hyperlattice and the orthogonal idempotnet elements form a quasi-distributive hyperboolean algebra.
TL;DR: In this paper, it was shown that aE + bF is Fredholm if and only if E + F is, thus answering affirmatively a question asked by Koliha and Rakocevic.
Abstract: Let E and F be idempotent operators on a complex Hilbert space, and let a and b be nonzero scalars with a + b ≠ 0. We prove that aE + bF is Fredholm if and only if E + F is, thus answering affirmatively a question asked by Koliha and Rakocevic.
Posted Content•
TL;DR: It is shown that among all tight designs in FP, only 5-designs in CP have irrational angle set.
Abstract: It is shown that among all tight designs in FP^n, where F is R, C, or H (quaternions), other than RP^1, only 5-designs in CP^1 have irrational angle set. This is the only case of equal ranks of the first and the last irreducible idempotent in the corresponding Bose-Mesner algebra.
Journal Article•
TL;DR: In this article, a simplicial idempotent augmented endof-unctor was constructed under the assumption of Vopenka's principle, such that the cellularization with respect to the cofibrant objects in the simplicial combinatorial model category was obtained.
Abstract: Given a simplicial idempotent augmented endofunctor $F$ on a simplicial combinatorial model category $M$, under the assumption of Vopenka's principle, we exhibit a set $A$ of cofibrant objects in $M$ such that $F$ is equivalent to $\CW_A$, the cellularization with respect to $A$.
TL;DR: In this article, conditions on the scalars defining a plenary train algebra of rank 4 were studied to assure the existence of such an idempotent element, which is an open problem to be solved.
Abstract: The existence of idempotent elements in plenary train algebras of rank greater than 3, is an open problem to be solved. J. Carlos Gutierrez's results on plenary train algebras in Gutierrez (2000) are based on the underlying assumption of the existence of an idempotent. In this article we study conditions on the scalars defining a plenary train algebra of rank 4 to assure the existence of such an idempotent.
01 Jan 2007
TL;DR: In this paper, the authors characterized the bijective linear preservers of idempotence on Tn(F) and the strong linear presers of idemepotence over Tn (F) are characterized.
Abstract: Let F be any field and let Tn(F) be the n × n upper triangular matrix space over F. We denote the set of all n × n upper triangular idempotent matrices over F by Pn(F). A map ϕ on Tn(F) is called a preserver of idempotence if ϕ(Pn(F)) ⊂ Pn(F); and a strong preserver of idempotence if ϕ(Pn(F)) = Pn(F). In this paper, we characterize the bijective linear preservers of idempotence on Tn(F). Further, the strong linear preservers of idempotence on Tn(F) are characterized. Mathematics Subject Classifications: 15A04; 15A03
TL;DR: In this paper, the properties of *-weakly regular modules, a generalization of fully idempotent rings, have been studied, which is a generalisation of regular modules by Ramamurthy and Mabuchi.
Abstract: We study generalizations of regular modules by Ramamurthy and Mabuchi. These are also generalizations of fully right idempotent and fully left idempotent rings, respectively. We also define and study the properties of *-weakly regular modules, a generalization of fully idempotent rings.
TL;DR: In this article, the authors give bounds on the distance from a non-zero idempotent to the set of nilpotents in a set of n × n matrices.
Abstract: We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of n × n matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.
TL;DR: In this paper, a survey of known theorems on bijective maps on idempotent operators preserving order or orthogonality is presented. And a new relation on the set of bounded linear idemomorphists on a Banach space X is shown.
Abstract: The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P . Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will be explained. At the end we will prove a new result concerning bijective maps on idempotent operators preserving comparability.
01 Jan 2007
TL;DR: In this article, it was shown that there is a one-to-one correspondence between bounded half-lattices and commutative idempotent monoids, and that the law of action and reaction is not an independent axiom but a consequence of fundamental balance laws.
Abstract: In this paper, the interplay between certain mathematical structures is elucidated. First, it is shown that there is a one-to-one correspondence between bounded half-lattices and commutative idempotent monoids (c.i.-monoids). Adding certain additional structural ingredients and axioms, such c.i.-momoids become Boolean algebras. There is a non-trivial one-to-one correspondence between these and what we call materially ordered sets, which are half -lattices that satisfy certain additional axioms. Such materially ordered sets can serve as mathematical models for certain physical systems. The correspondence between materially ordered sets and Boolean algebras can be used to show, for example, that the law of action and reaction (Newton’s third law) is not an independent axiom but a consequence of fundamental balance laws. 0. Mathematical Structures A mathematical structure is described by prescribing ingredients and postulating axioms, which are conditions that the ingredients are assumed to satisfy. In most cases, one starts with a single set and endows it with structure by specifying ingredients that are entities involving constructions from this given set. An isomorphism between two structures of the same type is an invertible mapping between the underlying sets that induces a correspondence between the ingredients. An automorphism is an isomorphism from the structured set to itself. Given a set S endowed with a specified structure and an arbitrary invertible mapping from S to a set T , one can use this mapping to transport the structure from S to T by transporting the ingredients of S to T . The axioms for T are then automatically satisfied. In some of the cases, the set T may coincide with S and then S acquires a second structure of the same type. The mapping is an automorphism only if this second structure coincides with the given one. These considerations will be illustrated by the content of the remainder of this paper. Given a set S we define Sub S to be the set of all subsets of S. Let f : A → B be a mapping with domain A and codomain B. The image mapping of f is the mapping f> : Sub A → Sub B defined by f>(U) := {f(x) | x ∈ U} for all U ∈ Sub A. (1) Let S ∈ Sub A and T ∈ Sub B be such that f>(S) ⊆ T . Then the adjustment f |S : S → T of f is defined by f |S (x) := f(x) for all x ∈ S. (2) A pre-monoid is a set M endowed with structure by the prescription of a mapping cmb : M×M → M called combination, which satisfies the associative axiom cmb(cmb(a, b), c) = cmb(a, cmb(b, c)) for all a, b, c ∈ M. (3) Amonoid M is a pre-monoid, with combination cmb, endowed with additional structure by the prescription of a neutral nt ∈ M which satisfies the neutrality axiom cmb(a,nt) = cmb(nt, a) = a for all a ∈ M. (4)
TL;DR: In this paper, it was shown that every locally idempotent (locally m-pseudoconvex) Hausdor algebra A with pseudoconvex vonNeumann bornology is a regular (respectively, bornological) inductive limit of metrizable locally m-(kB-convex)-subalgebras AB of AB of A.
Abstract: It is shown that every locally idempotent (locally m-pseudoconvex) Hausdor algebra A with pseudoconvex vonNeumannbornologyis a regular (respectively, bornological) inductive limit of metrizable locally m-(kB-convex) subalgebras AB of A In the case where A, in addition, is sequentially BA-complete (sequentially advertibly complete), then every subalgebraAB is a locally m-(kB-convex) Frechet algebra (respectively, an advertibly complete metrizable locally m-(kB-convex) algebra) for some kB 2 (0,1) Moreover, for a commutative unital locally m-pseudoconvex Hausdor algebra A over C with pseudoconvex von Neumann bornology, which at the same time is sequentially BA-complete and advertibly complete, the statements (a)-(j) of Proposition 32 are equivalent