Showing papers on "Idempotence published in 2017"
TL;DR: The ideals of the partition, Brauer, and Jones monoid are studied, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham--Houghton graphs, which relate to several well-studied topics in graph theory including the problem of counting perfect matchings.
45 citations
TL;DR: In this article, the authors studied the structure of the partial Brauer monoid and its planar sub-monoid, the Motzkin monoid, and obtained necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated.
43 citations
TL;DR: In this article, a single nontrivial equational condition which is implied by any non-nontrivial idempotent non-equational condition has been found, and the condition can be expressed in terms of terms that meet the required equations.
Abstract: An equational condition is a set of equations in an algebraic language, and an algebraic structure satisfies such a condition if it possesses terms that meet the required equations. We find a single nontrivial equational condition which is implied by any nontrivial idempotent equational condition.
29 citations
TL;DR: In this paper, it was shown that 3-transposition groups can be algebraically characterised as Matsuo algebras with idempotents satisfying the fusion rule Φ ( α ) for some α.
23 citations
TL;DR: In this paper, it was shown that an algebra satisfying mild additional assumptions can be realized as the endomorphism algebra of a cluster-tilting object in a Frobenius category, and if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi-Yau with respect to its frozen idempotent.
Abstract: We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of \((d-1)\)-cluster-tilting objects in certain stably \((d-1)\)-Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a \((d-1)\)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger ‘bimodule’ internally d-Calabi–Yau condition, this Frobenius category is stably \((d-1)\)-Calabi–Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi–Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot’s construction in the coefficient-free case.
18 citations
15 May 2017
TL;DR: A finitely-based variety of cyclic involutive GBI-algebras are constructed from so-called weakening relations, and it is proved that the class of weakening relation algebrAs is not finitely axiomatizable.
Abstract: This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Langer, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening relation algebras is not finitely axiomatizable. These algebras play a role similar to representable relation algebras, and we identify a finitely-based variety of cyclic involutive GBI-algebras that includes all weakening relation algebras. We also show that algebras of down-closed sets of partially-ordered groupoids are bounded cyclic involutive GBI-algebras.
17 citations
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TL;DR: In this article, it was shown that Brauer graph algebras coincide with the class of indecomposable idempotent biserial weighted surface algesbras.
Abstract: We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented triangles, investigated in [17] and [18]. Moreover we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in [17] and [19].
17 citations
TL;DR: In this paper, the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid were characterized in terms of combinatorial data associated with certain normal forms.
Abstract: We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.
13 citations
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TL;DR: In this paper, the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general IDempotent decompositions of identity, was introduced.
Abstract: In these notes we develop some basic theory of idempotents in monoidal categories. We introduce and study the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general idempotent decompositions of identity. If $\mathbf{E}$ is a categorical idempotent then $\operatorname{End}(\mathbf{E})$ is a graded commutative algebra. The same is true of $\operatorname{Hom}(\mathbf{E},\mathbf{E}^c[1])$ under certain circumstances, where $\mathbf{E}^c$ is the complement. These generalize the notions of cohomology and Tate cohomology of a finite dimensional Hopf algebra, respectively.
13 citations
TL;DR: In this article, the conservative algebra W(n) of all algebras on the n-dimensional vector space was defined, and automorphisms, one-sided ideals, and idempotents of W(2) were described.
Abstract: In 1990 Kantor defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n>1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of W(2). Also similar problems are solved for the algebra W2 of all commutative algebras on the 2-dimensional vector space and for the algebra S2 of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.
13 citations
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TL;DR: In this paper, the theory of categorical diagonalization is studied for Hecke algebras of type A. In particular, the authors construct a Young symmetrizer by simultaneously diagonalizing certain functors associated to the full twist braids.
Abstract: This paper lays the groundwork for the theory of categorical diagonalization Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each eigenspace These idempotents are mutually orthogonal and sum to the identity We categorify these tools At the categorical level, one has not only eigenobjects and eigenvalues but also eigenmaps, which relate an endofunctor to its eigenvalues Given an invertible endofunctor of a triangulated category with a sufficiently nice collection of eigenmaps, we construct idempotent functors which project to eigencategories These idempotent functors are mutually orthogonal, and a convolution thereof is isomorphic to the identity functor In several sequels to this paper, we will use this technology to study the categorical representation theory of Hecke algebras In particular, for Hecke algebras of type A, we will construct categorified Young symmetrizers by simultaneously diagonalizing certain functors associated to the full twist braids
TL;DR: In this article, it was shown that a ring R is strongly nil clean if and only if for any a, ∈ R, there exists an idempotent e ∈ ǫℤ[a] such that a − e 2k∈ N(R) for fixed m,n ∈ n,n∈,ℕ, a−a 2n+2m(n+1)∈ n(R), if and if for fixed n, n ∈,n,∈,ℚ,n
Abstract: A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a ∈ R, there exists an idempotent e ∈ ℤ[a] such that a − e ∈ N(R), if and only if R is periodic and R∕J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove that R is strongly nil clean if and only if for all a ∈ R, there exist n ∈ ℕ,k ≥ 0 (depending on a) such that an−an+2k∈N(R), if and only if for fixed m,n ∈ ℕ, a−a2n+2m(n+1)∈N(R) for all a ∈ R. These also extend known theorems, e.g, [5, Theorem 3.21], [6, Theorem 3], [7, Theorem 2.7] and [12, Theorem 2].
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TL;DR: In this paper, the authors show how to reduce the computation of the Eulerian idempotent to a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees.
Abstract: The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to establish explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The problem of finding formulas for the pre-Lie logarithm, which is interesting in its own right -- being related to operad theory, numerical analysis and renormalization -- is addressed using techniques inspired by umbral calculus. As a consequence of our analysis, we find formulas both for the Eulerian idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.
TL;DR: This computational method, together with interpolative aggregation, can be used for the development of general idempotent logic aggregators that satisfy a variety of conditions necessary for building decision models in the area of weighted compensative logic.
Abstract: We propose weighted aggregation algorithms for creating general idempotent weighted aggregators of n variables derived from related symmetric idempotent aggregators of two variables. This computational method, together with interpolative aggregation, can be used for the development of general idempotent logic aggregators that satisfy a variety of conditions necessary for building decision models in the area of weighted compensative logic.
TL;DR: In this article, the authors use a certain extension Sc(L) of a frame L, which is a Boolean and idempotent extension of a Frame L and show that it yields the desired results in the treatment of semicontinuity.
Abstract: Point-free modeling of mappings that are not necessarily continuous has been so far based on the extension of a frame to its frame of sublocales, mimicking the replacement of a topological space by its discretization. This otherwise successful procedure has, however, certain disadvantages making it not quite parallel with the classical theory (see Introduction). We mend it in this paper using a certain extension Sc(L) of a frame L, which is, a.o., Boolean and idempotent. Doing this we do not lose the merits of the previous approach. In particular we show that it yields the desired results in the treatment of semicontinuity. Also, there is no obstacle to using it as a basis of a point-free theory of rings of real functions; the \ring of all real functions" F(L) = C(Sc(L)) is now order complete. Mathematics Subject Classication (2010): 06D22, 54C30.Key words: Frame, locale, subt, regular, sublocale, sublocale lattice, open sublocale,closed sublocale, real function, lower and upper semicontinuities, lower and upper regularizations.
19 Jul 2017
TL;DR: A method for efficiently checking idempotence by combining the testing and static verification approaches is presented, which dramatically decreases the number of test cases used to check code including external scripts by applying the static verification approach.
Abstract: Infrastructure as Code, which uses machine-processable code for managing, provisioning, and configuring computing infrastructure, has been attracting wide attention. In its application, the idempotence of the code is essential: the system should converge to the desired state even if the code is repeatedly executed possibly with failures or interruptions. Previous studies have used testing or static verification techniques to check whether the code is idempotent or not. The testing approach is impractically time-consuming, whereas the static verification approach is not applicable in many practical cases in which external scripts are used. In this paper, we present a method for efficiently checking idempotence by combining the testing and static verification approaches. The method dramatically decreases the number of test cases used to check code including external scripts by applying the static verification approach.
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TL;DR: The Zariski topology or the intuitionistic fuzzy prime spectrum of a ring is introduced and it is shown that this topology is always T0-space and is T1-space when R is a ring in which every prime ideal is maximal, but even in this case it is not T2-space.
Abstract: In this paper, we have introduced the topological structure on the set of all intuitionistic fuzzy prime ideals of a ring. This topology is called the Zariski topology or the intuitionistic fuzzy prime spectrum of a ring. We have shown that this topology is always T0-space and is T1-space when R is a ring in which every prime ideal is maximal, but even in this case it is not T2-space. We have also studied a special subspace Y which is always compact and is connected if and only if 0 and 1 are the only idempotent in R. We have also shown that, when the ring R is Boolean ring, then the subspace Y is also T2 – space. An embedding of space X ¢ onto a subspace X* = {A I X | A is f –invariant} has been established.
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TL;DR: In this article, a finite-dimensional Jacobian algebra of a self-injective quiver with potential is constructed from a Postnikov diagram in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent.
Abstract: We study a finite-dimensional algebra $\Lambda$ constructed from a Postnikov diagram $D$ in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus $\Lambda$ is isomorphic to the stable endomorphism algebra of the cluster tilting module $T\in\underline{\operatorname{CM}}(B)$ introduced by Jensen-King-Su in order to categorify the cluster algebra structure of $\mathbb C[\operatorname{Gr}_k(\mathbb C^n)]$. We show that $\Lambda$ is self-injective if and only if $D$ has a certain rotational symmetry. In this case, $\Lambda$ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.
TL;DR: In this paper, tropical algebra is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models, which highlights an underlying order relation that is explored through the concepts of filter and ideal.
Abstract: Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents (elements) that is well-behaved with respect to composition. This kind of connection is studied with maps that preserve a monoid structure. The approach highlights an underlying order relation that is explored through the concepts of filter and ideal. Particular attention is paid to asymmetry and duality between max- and min-criteria. Physical implementations are presented through simple examples in thermodynamics and non-equilibrium physics. The phenomenon of ultrametricity, the notion of tropical equilibrium and the role of ground energy in non-equilibrium models are discussed. Tropical symmetry, i.e. idempotence, is investigated.
12 Jun 2017
TL;DR: It is shown that there exists unique idempotent nullnorm on an arbitrary distributive bounded lattice, and it is proved that an idem Potentnullnorm may not always exist on every bounded lattices.
Abstract: Nullnorms are generalizations of triangular norms (t-norms) and triangular conorms (t-conorms) with a zero element to be an arbitrary point from an arbitrary bounded lattice. In this paper, we study on the existence of idempotent nullnorms on bounded lattices. We show that there exists unique idempotent nullnorm on an arbitrary distributive bounded lattice. We prove that an idempotent nullnorm may not always exist on every bounded lattice. Furthermore, we propose the construction method to obtain idempotent nullnorms on a bounded lattice under additional assumptions on given zero element. As by-product of this method, we see that it is in existence an idempotent nullnorm on non-distributive bounded lattices.
TL;DR: In this paper, the question of whether an idempotent can be proven to split in the presence of propositional truncation and Voevodsky's univalence axiom is investigated.
Abstract: We study idempotents in intensional Martin-L\"of type theory, and in
particular the question of when and whether they split. We show that in the
presence of propositional truncation and Voevodsky's univalence axiom, there
exist idempotents that do not split; thus in plain MLTT not all idempotents can
be proven to split. On the other hand, assuming only function extensionality,
an idempotent can be split if and only if its witness of idempotency satisfies
one extra coherence condition. Both proofs are inspired by parallel results of
Lurie in higher category theory, showing that ideas from higher category theory
and homotopy theory can have applications even in ordinary MLTT.
Finally, we show that although the witness of idempotency can be recovered
from a splitting, the one extra coherence condition cannot in general; and we
construct "the type of fully coherent idempotents", by splitting an idempotent
on the type of partially coherent ones. Our results have been formally verified
in the proof assistant Coq.
TL;DR: In this paper, the sub-power membership problem (SMP) was shown to be NP-complete for finite semigroups and the greatest variety of bands all of whose finite members induce a tractable SMP was determined.
TL;DR: In this paper, it was shown that the K-theory of C*-algebras can be defined by pairs of matrices satisfying less strict relations than idempotency.
Abstract: We show that the K-theory of C*-algebras can be defined by pairs of matrices satisfying less strict relations than idempotency.
TL;DR: Topological self-similarity in complex networks representing diverse forms of connectivity in the brain and some related dynamical systems is investigated, by considering the correlation between edges directly connecting any two nodes in a network and indirect connection between the same via all triangles spanning the rest of the network.
Abstract: Self-similarity across length scales is pervasively observed in natural systems. Here, we investigate topological self-similarity in complex networks representing diverse forms of connectivity in the brain and some related dynamical systems, by considering the correlation between edges directly connecting any two nodes in a network and indirect connection between the same via all triangles spanning the rest of the network. We note that this aspect of self-similarity, which is distinct from hierarchically nested connectivity (coarse-grain similarity), is closely related to idempotence of the matrix representing the graph. We introduce two measures, ι(1) and ι(∞), which represent the element-wise correlation coefficients between the initial matrix and the ones obtained after squaring it once or infinitely many times, and term the matrices which yield large values of these parameters "quasi-idempotent". These measures delineate qualitatively different forms of "shallow" and "deep" quasi-idempotence, which are influenced by nodal strength heterogeneity. A high degree of quasi-idempotence was observed for partially synchronized mean-field Kuramoto oscillators with noise, electronic chaotic oscillators, and cultures of dissociated neurons, wherein the expression of quasi-idempotence correlated strongly with network maturity. Quasi-idempotence was also detected for macro-scale brain networks representing axonal connectivity, synchronization of slow activity fluctuations during idleness, and co-activation across experimental tasks, and preliminary data indicated that quasi-idempotence of structural connectivity may decrease with ageing. This initial study highlights that the form of network self-similarity indexed by quasi-idempotence is detectable in diverse dynamical systems, and draws attention to it as a possible basis for measures representing network "collectivity" and pattern formation.
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TL;DR: In this article, it was shown that Olsak's equation systems can't be compressed into just one equation, and that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms.
Abstract: It is known that any finite idempotent algebra that satisfies a nontrivial Maltsev condition must satisfy the linear one-equality Maltsev condition (a variant of the term discovered by M. Siggers and refined by K. Kearnes, P. Markovic, and R. McKenzie):
\[
t(r,a,r,e)\approx t(a,r,e,a).
\] We show that if we drop the finiteness assumption, the $k$-ary weak near unanimity equations imply only trivial linear one-equality Maltsev conditions for every $k\geq 3$. From this it follows that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms.
Miroslav Olsak has recently shown that there is a weakest nontrivial strong Maltsev condition for idempotent algebras. Olsak has found several such (mutually equivalent) conditions consisting of two or more equations. Our result shows that Olsak's equation systems can't be compressed into just one equation.
13 Jun 2017
TL;DR: The results show that for a \(\mathcal K\)-valued formal context (G, M, R)—where \(|G| = g\), \(|M| = m\) and \(R \in K^{g\times {m}}\)—there are only two different “shapes” of lattices, suggesting a notion of a 4-concept associated to a formal concept.
Abstract: We report on progress relating \(\mathcal K\)-valued FCA to \(\mathcal K\)-Linear Algebra where \(\mathcal K\) is an idempotent semifield We first find that the standard machinery of linear algebra points to Galois adjunctions as the preferred construction, which generates either Neighbourhood Lattices of attributes or objects For the Neighbourhood of objects we provide the adjoints, their respective closure and interior operators and the general structure of the lattices, both of objects and attributes Next, these results and those previous on Galois connections are set against the backdrop of Extended Formal Concept Analysis Our results show that for a \(\mathcal K\)-valued formal context (G, M, R)—where \(|G| = g\), \(|M| = m\) and \(R \in K^{g\times {m}}\)—there are only two different “shapes” of lattices each of which comes in four different “colours”, suggesting a notion of a 4-concept associated to a formal concept Finally, we draw some conclusions as to the use of these as data exploration constructs, allowing many different “readings” on the contextualized data
TL;DR: In this article, the maximal regular subsemigroups of when is a finite-dimensional subspace of over a finite field were derived and the rank and idempotent rank for when is an -dimensional subspaces of an -deterministic vector space over a fixed field.
Abstract: Let be a vector space and let denote the semigroup (under composition) of all linear transformations from into . For a fixed subspace of , let be the semigroup consisting of all linear transformations from into . In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc. 77(3) (2008), 441–453] proved that is the largest regular subsemigroup of and characterized Green’s relations on . In this paper, we determine all the maximal regular subsemigroups of when is a finite-dimensional subspace of over a finite field. Moreover, we compute the rank and idempotent rank of when is an -dimensional subspace of an -dimensional vector space over a finite field .
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TL;DR: In this paper, the Peirce decomposition induced by an idempotent provides a natural environment for defining and classifying new types of rings, and this point of view offers a way to unify and to expand the classical theory of semiperfect rings and idempots to much larger classes of rings.
Abstract: Idempotents dominate the structure theory of rings. The Peirce decomposition induced by an idempotent provides a natural environment for defining and classifying new types of rings. This point of view offers a way to unify and to expand the classical theory of semiperfect rings and idempotents to much larger classes of rings. Examples and applications are included.
TL;DR: In this paper, it was shown that a finite dimensional algebra A over a field K is gendo-symmetric if and only if there is a bocs-structure on ( A, D ( A ), where D = H o m K ( −, K ) is the natural duality.
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TL;DR: In particular, this paper showed that if a ring is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1e)$ are both strongly nil-clean rings, then $R/J(R)$ is also a strongly non-clean ring.
Abstract: We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings, then $R/J(R)$ is nil-clean. In particular, under certain additional circumstances, $R$ is also nil-clean. These results somewhat improves on achievements due to Diesl in J. Algebra (2013) and to Koc{s}an-Wang-Zhou in J. Pure Appl. Algebra (2016). In addition, we also give a new transparent proof of the main result of Breaz-Calugareanu-Danchev-Micu in Linear Algebra Appl. (2013) which says that if $R$ is a commutative nil-clean ring, then the full $ntimes n$ matrix ring $mathbb{M}_n(R)$ is nil-clean.