Showing papers on "Idempotence published in 2012"

Idempotence is not a medical condition

Abstract: The definition of distributed computing can be confusing. Sometimes, it refers to a tightly coupled cluster of computers working together to look like one larger computer. More often, however, it r...

Maximal subgroups of free idempotent-generated semigroups over the full transformation monoid

Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n− 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr. 2000 Mathematics Subject Classification: 20M05, 05E15, 20F05.

Finitely Related Clones and Algebras with Cube Terms

Abstract: Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related—every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.

Fully idempotent and coidempotent modules

Abstract: In this paper, the notion of fully idempotent modules is dened and it is shown that this notion inherits most of the essential properties of the usual notion of von Neumann's regular rings. Furthermore, we introduce the dual notion of fully idempo- tent modules (that is, fully coidempotent modules) and investigate some properties of this class of modules.

Additive mappings derivable at non-trivial idempotents on Banach algebras

Abstract: Let 𝒜 be a Banach algebra with unity I containing a non-trivial idempotent P and ℳ be a unital 𝒜-bimodule. Under several conditions on 𝒜, ℳ and P, we show that if d : 𝒜 → ℳ is an additive mapping derivable at P (i.e. d(AB) = Ad(B) + d(A)B for any A, B ∈ 𝒜 with AB = P), then d is a derivation or d(A) = τ(A) + AN for some additive derivation τ : 𝒜 → ℳ and some N ∈ ℳ, and various examples are given which illustrate limitations on extending some of the theory developed. Also, we describe the additive mappings derivable at P on semiprime Banach algebras and C*-algebras. As applications of the above results, we characterize the additive mappings derivable at P on matrix algebras, Banach space nest algebras, standard operator algebras and nest subalgebras of von Neumann algebras. Moreover, we obtain some results about automatic continuity of linear (additive) mappings derivable at P on various Banach algebras.

Special Dual Like Numbers and Lattices, by W. B. Vasantha Kandasamy, Florentin Smarandache

Abstract: In this book we define x = a+bg to be a special dual like number, where a, b are reals and g is a new element such that g^2 = g. The new element which is idempotent can be got from Z_n or from lattices or from linear operators. Mixed dual numbers are constructed using dual numbers and special dual like numbers. Neutrosophic numbers are a natural source of special dual like numbers, since they have the form a+bI, where I = indeterminate and I^2 = I.

Dynamics of two-dimensional evolution algebras

Abstract: Recently by Casas, Ladra and Rozikov a notion of a chain of evolution algebras is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies the Chapman-Kolmogorov equation. In this paper we construct 25 distinct examples of chains of two-dimensional evolution algebras. For all of these 25 chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

An algebraic approach to multidimensional minimax location problems with Chebyshev distance

Abstract: Minimax single facility location problems in multidimensional space with Chebyshev distance are examined within the framework of idempotent algebra. The aim of the study is twofold: first, to give a new algebraic solution to the location problems, and second, to extend the area of application of idempotent algebra. A new algebraic approach based on investigation of extremal properties of eigenvalues for irreducible matrices is developed to solve multidimensional problems that involve minimization of functionals defined on idempotent vector semimodules. Furthermore, an unconstrained location problem is considered and then represented in the idempotent algebra settings. A new algebraic solution is given that reduces the problem to evaluation of the eigenvalue and eigenvectors of an appropriate matrix. Finally, the solution is extended to solve a constrained location problem.

Rational group algebras of finite groups: from idempotents to units of integral group rings

Abstract: We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring ${\mathbb Z} G$ of a finite nilpotent group G has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups (H, K), called strong Shoda pairs, and explicit constructed central elements e(G, H, K) play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such e(G, H, K), with (H, K) strong Shoda pairs in subgroups of G.

A note on maximal subgroups of free idempotent generated semigroups over bands

Abstract: We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.

The join levels of the trotter-weil hierarchy are decidable

Abstract: The variety DA of finite monoids has a huge number of different characterizations, ranging from two-variable first-order logic FO2 to unambiguous polynomials. In order to study the structure of the subvarieties of DA, Trotter and Weil considered the intersection of varieties of finite monoids with bands, i.e., with idempotent monoids. The varieties of idempotent monoids are very well understood and fully classified. Trotter and Weil showed that for every band variety V there exists a unique maximal variety W inside DA such that the intersection with bands yields the given band variety V. These maximal varieties W define the Trotter-Weil hierarchy. This hierarchy is infinite and it exhausts DA; induced by band varieties, it naturally has a zigzag shape. In their paper, Trotter and Weil have shown that the corners and the intersection levels of this hierarchy are decidable. In this paper, we give a single identity of omega-terms for every join level of the Trotter-Weil hierarchy; this yields decidability. Moreover, we show that the join levels and the subsequent intersection levels do not coincide. Almeida and Azevedo have shown that the join of $\mathcal R$-trivial and $\mathcal L$-trivial finite monoids is decidable; this is the first non-trivial join level of the Trotter-Weil hierarchy. We extend this result to the other join levels of the Trotter-Weil hierarchy. At the end of the paper, we give two applications. First, we show that the hierarchy of deterministic and codeterministic products is decidable. And second, we show that the direction alternation depth of unambiguous interval logic is decidable.

Idempotent elements of the endomorphism semiring of a finite chain

Abstract: Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well known semiring. The construction of an equivalence relation such that any equivalence class contain just one idempotent is proposed. In our main result we prove that such equivalence class is a semiring and find his order. We prove that the set of all idempotents with certain jump points is a semiring.

A solution of a tropical linear vector equation

Abstract: A linear vector equation is considered defined in terms of idempotent mathematics. To solve the equation, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces and reduces the solution of the equation to that of a tropical optimization problem. Based on the approach, existence and uniqueness conditions are established for the solution, and a general solution to the equation is given.

Geometrical aspects of possibility measures on finite domain MV-clans

Abstract: In this paper, we study generalized possibility and necessity measures on MV-algebras of [0, 1]-valued functions (MV-clans) in the framework of idempotent mathematics, where the usual field of reals $${\mathbb{R}}$$ is replaced by the max-plus semiring $${\mathbb{R}}_{\rm max}.$$ We prove results about extendability of partial assessments to possibility and necessity measures, and characterize the geometrical properties of the space of homogeneous possibility measures. The aim of the present paper is also to support the idea that idempotent mathematics is the natural framework to develop the theory of possibility and necessity measures, in the same way classical mathematics serves as a natural setting for probability theory.

The Join Levels of the Trotter-Weil Hierarchy are Decidable

Abstract: The variety DA of finite monoids has a huge number of different characterizations, ranging from two-variable first-order logic FO^2 to unambiguous polynomials. In order to study the structure of the subvarieties of DA, Trotter and Weil considered the intersection of varieties of finite monoids with bands, i.e., with idempotent monoids. The varieties of idempotent monoids are very well understood and fully classified. Trotter and Weil showed that for every band variety V there exists a unique maximal variety W inside DA such that the intersection with bands yields the given band variety V. These maximal varieties W define the Trotter-Weil hierarchy. This hierarchy is infinite and it exhausts DA; induced by band varieties, it naturally has a zigzag shape. In their paper, Trotter and Weil have shown that the corners and the intersection levels of this hierarchy are decidable. In this paper, we give a single identity of omega-terms for every join level of the Trotter-Weil hierarchy; this yields decidability. Moreover, we show that the join levels and the subsequent intersection levels do not coincide. Almeida and Azevedo have shown that the join of R-trivial and L-trivial finite monoids is decidable; this is the first non-trivial join level of the Trotter-Weil hierarchy. We extend this result to the other join levels of the Trotter-Weil hierarchy. At the end of the paper, we give two applications. First, we show that the hierarchy of deterministic and codeterministic products is decidable. And second, we show that the direction alternation depth of unambiguous interval logic is decidable.

On tropical Kleene star matrices and alcoved polytopes

Abstract: In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero--diagonal real matrix $A$ is characterized by $A$ being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.

The subvariety lattice for representable idempotent commutative residuated lattices

Abstract: RICRL denotes the variety of commutative residuated lattices which have an idempotent monoid operation and are representable in the sense that they are subdirect products of linearly ordered algebras. It is shown that the subvariety lattice of RICRL is countable, despite its complexity and in contrast to several varieties of closely related algebras.

Projective bichains

Abstract: An algebra with two binary operations · and + that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) = x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation + and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems.

Primitive Central Idempotents in Rational Group Algebras

Abstract: A complex irreducible character χ of a finite group G, with an affording representation ρ, is defined to have the property 𝒫 if, for all g ∈ G, either χ(g) = 0 or all the eigen-values of ρ(g) have the same order. An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] associated with a complex irreducible character having the property 𝒫 is derived. Several consequences are then obtained.

Dequantization of Mathematics, idempotent semirings and fuzzy sets

Abstract: In this talk a brief survey of basic ideas of Idempotent Mathematics is presented. Relations between this theory and the theory of fuzzy sets as well as the possibility theory and some applications (including computer applications) are discussed.