Showing papers on "Idempotence published in 2021"
TL;DR: The 6-element Brandt monoid $$B_2^1$$ admits a unique addition under which it becomes an additively idempotent semiring as discussed by the authors, and it has been shown that this addition is a term operation of the monoid as an inverse semigroup.
Abstract: The 6-element Brandt monoid $$B_2^1$$
admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of $$B_2^1$$
as an inverse semigroup. As a consequence, we exhibit an easy proof that the semiring identities of $$B_2^1$$
are not finitely based.
10 citations
TL;DR: In this article, the authors studied a bijective linear map f:Mn(C)→Mn (C) satisfying the property f(X), where X is a fixed rank-one idempotents of Mn(C), the algebra of n×n matrices over the complex numbers.
Abstract: Let P, Q be fixed rank-one idempotents of Mn(C), the algebra of n×n matrices over the complex numbers. In this paper, we will study a bijective linear map f:Mn(C)→Mn(C) satisfying the property f(X)...
9 citations
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TL;DR: In this paper, the authors define a higher unitary idempotent completion for C* 2-categories called Q-system completion and study its properties, showing that the C * 2-category of right correspondences of unital C*-algebras is Q- system complete by constructing an inverse realization $†$ 2-functor.
Abstract: A Q-system in a C* 2-category is a unitary version of a separable Frobenius algebra object and can be viewed as a unitary version of a higher idempotent. We define a higher unitary idempotent completion for C* 2-categories called Q-system completion and study its properties. We show that the C* 2-category of right correspondences of unital C*-algebras is Q-system complete by constructing an inverse realization $†$ 2-functor. We use this result to construct induced actions of group theoretical unitary fusion categories on continuous trace C*-algebras with connected spectra.
6 citations
6 citations
01 Aug 2021
TL;DR: In this paper, it was shown that idempotent graphs are weakly perfect and characterized the rings of an abelian Rickart ring R with connected complements, which is the case for the zero-divisor graph.
Abstract: The idempotent graph I(R) of a ring R is a graph with nontrivial idempotents of R as vertices, and two vertices are adjacent in I(R) if and only if their product is zero. In the present paper, we prove that idempotent graphs are weakly perfect. We characterize the rings whose idempotent graphs have connected complements. As an application, the idempotent graph of an abelian Rickart ring R is used to obtain the zero-divisor graph $$\Gamma (R)$$
of R.
5 citations
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TL;DR: In this article, a new approach to the Hutchinson-Barnsley theory for idempotent measures is presented, which is based on the embedding of the space of the IDM space to the fuzzy set.
Abstract: We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ., 10 (2018), 1, 172--178. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as greyscale images.
4 citations
TL;DR: It is shown that the natural partial order induced by any commutative, associative, idempotent function F corresponds to a meet semi-lattice and that F ( x, y ) can be expressed by the meet of x and y with respect to the naturalpartial order.
4 citations
TL;DR: Recently, a new type of generalized inverse called the n-strong Drazin inverse was introduced by Mosic in the setting of rings as mentioned in this paper, where the inverse is defined in terms of an element x ∈ R.
Abstract: Recently, a new type of generalized inverse called the n-strong Drazin inverse was introduced by Mosic in the setting of rings. Namely, let R be a ring and n be a positive integer, an element x ∈ R...
4 citations
18 Jul 2021
TL;DR: In this paper, a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings is introduced. But it assumes that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and the coefficients of the difference equation are not degenerated.
Abstract: We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism we can reduce the problem to find solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains. Utilizing existing solvers in this integral domain setting, we obtain a general solver where the components of the linear difference equations and the solutions can be taken from difference rings that are built e.g., by RΠΣ-extensions over ΠΣ-fields. This class of difference rings contains, e.g., nested sums and products, products over roots of unity and nested sums defined over such objects.
3 citations
TL;DR: In this article, the authors demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C.
3 citations
TL;DR: In this paper, it was shown that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel" by an application of the category theory.
Abstract: Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I 3 (X)\ X implies the metrizability of X.
TL;DR: This study proposes several construction methods to obtain a semi-t-operator from a given semi-T-conorm and semi- t-norm on bounded lattices with additional constraints and discusses the presence of idempotent semi- T-operators on bound lattices.
Abstract: Recently, Fang and Hu introduced the definition of semi-t-operators on bounded lattices. In this study, we propose several construction methods to obtain such a semi-t-operator from a given semi-t-conorm and semi-t-norm on bounded lattices. Furthermore, we discuss the presence of idempotent semi-t-operators on bounded lattices, and show several different methods for construction of idempotent semi-t-operators on bounded lattices with additional constraints.
TL;DR: In this paper, it was shown that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoids of germs arising from the standard action of the inverse semigroup on the space of idempotent filters.
TL;DR: In this article, the properties of the local closure function and the spaces defined by it using common ideals, like ideals of finite sets, countable sets, closed and discrete sets, scattered sets and nowhere dense sets, are investigated.
Abstract: The aim of this paper is to continue the work started in Pavlovic (Filomat 30(14):3725–3731, 2016) . We investigate further the properties of the local closure function and the spaces defined by it using common ideals, like ideals of finite sets, countable sets, closed and discrete sets, scattered sets and nowhere dense sets. Also, closure compatibility between the topology and the ideal, idempotency, and cases when the local closure of the whole space X is X or a proper subset of X, are closely investigated. In the case of closure compatibility and idempotency of the local closure function, the topology obtained by the local closure function is completely described.
TL;DR: In this paper, the authors describe all idempotent matrices with only zeros and ones on the diagonal in T(n,R) -the ring of n×n upper triangular matrices over R (n∈N).
Abstract: Let R be an associative ring with identity 1. We describe all idempotent matrices with only zeros and ones on the diagonal in T(n,R) – the ring of n×n upper triangular matrices over R (n∈N), and T(...
TL;DR: In this paper, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs).
Abstract: The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.
TL;DR: This paper explores some new results on the distributivity equations between overlap (grouping) functions and null-uninorms, which are the generalization of uninorms and nullnorms and gives the full characterization of any idempotent null- uninorm.
TL;DR: In this article, it was shown that there is a one-to-one correspondence between idempotent functions on a set of size n, complete exceptional sequences of linear radical square zero Nakayama algebras of rank n and rooted labeled forests with n nodes and height of at most one.
Abstract: We prove that there is a one-to-one correspondence between the following three sets: idempotent functions on a set of size n, complete exceptional sequences of linear radical square zero Nakayama algebras of rank n and rooted labeled forests with n nodes and height of at most one. Therefore, the number of exceptional sequences is given by the sum ∑j=1nnjjn−j.
TL;DR: In this article, the authors characterize bijective linear maps on Mn(ℂ) that preserve the square roots of an idempotent matrix (of any rank) and present them as a direct sum of a map preserving involutions.
Abstract: We characterize bijective linear maps on Mn(ℂ) that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions ...
15 Jan 2021
TL;DR: In this article, the authors extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semields, at the same time.
Abstract: Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯-four-fold Formal Concept Analysis (K¯-4FCA) where K¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-a-vis previous formal conceptual analyses and some affordances of their results.
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TL;DR: It is proved that for a number of important semirings, including min-max semirINGS, and the semiring of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic, and thus elementary equivalence implies isomorphism.
Abstract: We study the first-order axiomatisability of finite semiring interpretations or, equivalently, the question whether elementary equivalence and isomorphism coincide for valuations of atomic facts over a finite universe into a commutative semiring. Contrary to the classical case of Boolean semantics, where every finite structure can obviously be axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and strongly depends on the underlying semiring. We prove that for a number of important semirings, including min-max semirings, and the semirings of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic. The same is true for the polynomial semirings that are universal for the classes of absorptive, idempotent, and fully idempotent semirings, respectively. On the other side, we prove that for other, practically relevant, semirings such as the Viterby semiring, the tropical semiring, the natural semiring and the universal polynomial semiring N[X], all finite semiring interpretations are first-order axiomatisable (and thus elementary equivalence implies isomorphism), although some of the axiomatisations that we exhibit use an infinite set of axioms.
TL;DR: In this paper, it was shown that a unital subalgebra of M 3 (C ) is projection compressible if and only if it is idempotent compressible.
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TL;DR: In this article, the authors adapt the standard notion of bisimilarity for topological models, namely Topo-bisimilarity, to closure models and refine it for quasi-discrete closure models.
Abstract: Closure spaces are a generalisation of topological spaces obtained by removing the idempotence requirement on the closure operator. We adapt the standard notion of bisimilarity for topological models, namely Topo-bisimilarity, to closure models -- we call the resulting equivalence CM-bisimilarity -- and refine it for quasi-discrete closure models. We also define two additional notions of bisimilarity that are based on paths on space, namely Path-bisimilarity and Compatible Path-bisimilarity, CoPa-bisimilarity for short. The former expresses (unconditional) reachability, the latter refines it in a way that is reminishent of Stuttering Equivalence on transition systems. For each bisimilarity we provide a logical characterisation, using variants of the Spatial Logic for Closure Spaces (SLCS). We also address the issue of (space) minimisation via the three equivalences.
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TL;DR: In this paper, the authors present a one-sided version of idempotent completion called left $E$-completion, which is a variant of a small category called a constellation.
Abstract: Given a monoid $S$ with $E$ any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left $E$-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs $S,E$ for which this happens, and characterise those left restriction semigroups that arise as such left $E$-completions of their monoid of elements having domain $1$. As first applications, we decompose the left restriction semigroup of partial functions on the set $X$ and the right restriction semigroup of left total partitions on $X$ as left and right $E$-completions respectively of the transformation semigroup $T_X$ on $X$, and decompose the left restriction semigroup of binary relations on $X$ under demonic composition as a left $E$-completion of the left-total binary relations. In many cases, including these three examples, the construction embeds in a semigroup Zappa-Szep product.
TL;DR: In this article, the class of endomorphisms in the semiring UTMn(S) of upper triangular matrices over an additively idempotent semiring S is described.
Abstract: We give a description of the class of endomorphisms in the semiring UTMn(S) of upper triangular matrices over an additively idempotent semiring S. The endomorphisms α such that α(Eij), where Eij is...
TL;DR: This paper investigates two families of autodistributive aggregation operations defined on finite linearly ordered scales: one with a neutral element and the other under some partial smoothness-related conditions.
TL;DR: Two new construction methods of nullnorms derived from an arbitrary t-norm using a closure operator using aclosure operator on bounded lattices, where some sufficient and necessary conditions are considered, are proposed.
TL;DR: In this article, the authors studied the problem of determining whether a finite algebra with finitely many basic operations contains a cube term, and they gave both structural and algorithmic results.
Abstract: We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most N, where the number N depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on N that, in the special case of algebras with more than $$\left( {\begin{array}{c}|A|\\ 2\end{array}}\right) $$
basic operations, improves an earlier result of K. Kearnes and A. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a k-ary near unanimity operation if and only if it contains a k-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.
TL;DR: In this paper, the existence of infinite-dimensional pre-Hilbert absolute-valued algebras satisfying the identity ( x 2, y, x 2 ) = 0 was shown.
TL;DR: In this article, it was shown that 2-torsion-free simple right-symmetric superrings having a nontrivial idempotent and satisfying a superidentity (x, y, z) + (−1)z(x+y)(z, x, y) + −1x(y+z)(y, z, x) = 0 are associative.
Abstract: It is proved that 2-torsion-free simple right-symmetric superrings having a nontrivial idempotent and satisfying a superidentity (x, y, z) + (−1)z(x+y)(z, x, y) + (−1)x(y+z)(y, z, x) = 0 are associative. As a consequence, every simple finitedimensional (1, 1)-superalgebra with semisimple even part over an algebraically closed field of characteristic 0 is associative.