Showing papers on "Idempotence published in 2008"

Inverse semigroups and combinatorial C*-algebras

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 semigroupoids, 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 consequently 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.

Existence theorems for weakly symmetric operations

Abstract: A k-ary weak near-unanimity operation (or k-WNU) on A is an operation that satisfies the equations w(x, . . . x) ≈ x and w(y, x, . . . , x) ≈ w(x, y, . . . , x) ≈ · · · ≈ w(x, x, . . . , x, y) . If an algebra A has a k-NU (or a k-WNU) term operation, we say that A satisfies NU(k) (or WNU(k), respectively). Likewise, a variety is said to satisfy NU(k) (or WNU(k), respectively), it it has a k-variable term satisfying these equations. It has been conjectured that a finite idempotent algebra A has finite relational width if and only if V(A) (the variety generated by A) has meet semi-distributive congruence lattices. The concept of “finite relational width” arises in the theory of complexity of algorithms, in the algebraic study of constraint-satisfaction problems. Actually, there are several different definitions of this concept and it is not known if they are equivalent. One version of the concept and the conjecture mentioned above are due to B. Larose and L. Zadori [10]. The important family of varieties with meet semi-distributive congruence lattices has various known characterizations. There is a characterization by a certain Maltsev condition; also, it is known that a locally finite variety has this property iff it omits congruence covers of types 1 and 2 (defined in the tame congruence theory of D. Hobby, R. McKenzie [6]). E. Kiss showed that a finite idempotent algebra of relational width k must have an m-WNU term operation for every m ≥ k. E. Kiss and M. Valeriote then observed that a finite algebra with a k-WNU term operation, k > 1, must omit congruence covers of type 1. These observations led M. Valeriote to make two conjectures: any locally finite variety omits congruence covers of type 1 iff it satisfies WNU(k) for some k > 1; any locally finite variety has meet semi-distributive congruence lattices if and only if for some k, it satisfies WNU(m) for all m ≥ k. In this paper, we prove both of these conjectures of M. Valeriote. The family of locally finite varieties omitting type 1 is the largest family of locally finite varieties defined by a nontrivial idempotent Maltsev condition. For this

A quantitative version of the idempotent theorem in harmonic analysis

Abstract: Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure // G M(G) is said to be idempotent if /x*/i = //, or alternatively if j5 takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure \x is idempotent if and only if the set {7 G G : i2(7) = 1} belongs to the coset ring of G, that is to say we may write L i=i

Classification of finite congruence-simple semirings with zero

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.

Subgroups of free idempotent generated semigroups need not be free

Abstract: We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup.

Cyclic projectors and separation theorems in idempotent convex geometry

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. Here we establish a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. The proof of this theorem involves specific nonlinear operators, called here cyclic projectors on idempotent semimodules. These are analogues of the cyclic nearest-point projections known in convex analysis. We obtain a theorem that characterizes the spectrum of cyclic projectors on idempotent semimodules in terms of a suitable extension of Hilbert's projective metric. We also deduce as a corollary of our main results the idempotent analogue of Helly's theorem.

On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables

Abstract: Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non-commuting, then the product of any two aperiodic series is again aperiodic. We also show that if K is idempotent and the matrix monoids over K have a Burnside property (satisfied, e.g. by the tropical semiring), then the aperiodic and the star-free series coincide. This generalizes a classical result of Schutzenberger (Inf. Control 4:245–270, 1961) for aperiodic regular languages and subsumes a result of Guaiana et al. (Theor. Comput. Sci. 97:301–311, 1992) on aperiodic trace languages.

On the lattice of equational classes of Boolean functions and its closed intervals

Abstract: Let A be a finite set with | A |M2. The composition of two classes I and J of operations on A, is defined as the set of all composites f (g1 ...,gn ) with f ŒI and g1 ...,gn ŒJ . This binary operation gives a monoid structure to the set EA of all equational classes of operations on A. The set EA of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in EA. In the Boolean case | A |=2, the lattice EA contains uncountably many (2?0 ) equational classes, but only countably many of them are clones. The aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its “closed" intervals [ C 1, C 2], for idempotent classes C 1 and C 2.For | A |=2, we give a complete classification of all closed intervals [C 1, C 2] in terms of their size, and provide a simple, necessary and sufficient condition characterizing the uncountable closed intervals of EA.

Idempotents and regular elements of complete semigroups of binary relations

Abstract: As is known (see [5, 6]), idempotent elements of semigroups play an important role in the investigation of semigroups themselves. Moreover, it is known that any semigroup can be isomorphically embedded into some semigroup of binary relations on some nonempty set X. This has generated great interest in the investigation of idempotent binary relations. Different authors used different approaches to study the construction of these elements and obtained different answers to the question under consideration (see, e.g., [1–3, 7, 8]). In this work, first, we characterize the complete X-semilattices of unions D for which there exist idempotent binary relations such that the set of all their cuts to the elements of D coincides with the given semilattice. Then we give a description of the structure of these idempotent binary relations in the language of limiting elements of some subset of the semilattice D (see Lemma 1 and Theorem 5) and show how we can find all idempotents of the complete group of binary relations BX (D) defined by the complete X-semilattice of unions D (see Theorem 6 and Corollary 4). Moreover, with the help of chains of the semilattice D, we give a rule for constructing the semigroups of idempotent elements of the semigroup BX(D) (see Theorem 7). In Theorem 9, we give a description of the structure of all regular elements of the semigroup BX (D).

Varieties with Definable Factor Congruences

Abstract: We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are "definable" in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0&1 and Boolean Factor Congruences. We also obtain an explicit first order definition of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner it is preserved by taking direct products and direct factors. The main tool is the use of "central elements," which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

Periodic elements of the free idempotent generated semigroup on a biordered set

Abstract: We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup

Power clones and non-deterministic hypersubstitutions

Abstract: Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set Wτ(X) of all terms of type τ. If σ : {fi | i ∈ I} → Wτ(X) is a hypersubstitution and its canonical extension, then the set is a tree transformation where the original language and the image language are of the same type. Tree transformations of the type Tσ can be produced by tree transducers. Here Tσ is the graph of the function . Since the set of all hypersubstitutions of type r forms a semigroup with respect to the multiplication , semigroup properties influence the properties of tree transformations of the form Tσ. For instance, if σ is idempotent, the relation Tσ is transitive (see [1]). Non-deterministic tree transducers produce tree transformations which are not graphs of some functions. If such tree transformations have the form Tσ, then σ is no longer a function. Therefore, there is some interest to study non-deterministic hypersubstitutions. That means, there are operation symbols which have not only one term of the corresponding arity as image, but a set of such terms. To define the extensions of non-deterministic hypersubstitutions, we have to extent the superposition operations for terms to a superposition defined on sets of terms. Let be the power set of the set of all n-ary terms of type τ. Then we define a superposition operation and get a heterogeneous algebra (ℕ+ is the set of all positive natural numbers), which is called the power clone of type τ. We prove that the algebra satisfies the well-known clone axioms (C1), (C2), (C3), where (C1) is the superassociative law (see e.g. [5], [4]). It turns out that the extensions of non-deterministic hypersubstitutions are precisely those endomorphisms of the heterogeneous algebra which preserve unions of families of sets. As a consequence, to study tree transformations of the form Tσ, where σ is a non-deterministic hypersubstitution, one can use the structural properties of non-deterministic hypersubstitutions. Sets of terms of type τ are tree languages in the sense of [3] and the operations are operations on tree languages. In [3] also another kind of superposition of tree languages is introduced which generalizes the usual complex product of subsets of the universe of a semigroup. We show that the extensions of non-deterministic hypersubstitutions are not endomorphisms with respect to this kind of superposition.

Essential operations in centralizer clones

Abstract: For an algebra A and for any finite n ≥ 2, a homomorphism \(h:A^{n} \rightarrow A\) is essentially n-ary if it depends on all its variables. The existence of such an h brings the integer n into the set S(cA) of all significant arities of the centralizer clone cA of the algebra A. For idempotent algebras, algebras with zero and congruence distributive algebras, the set S(cA) must be an order ideal in ω = {0, 1, . . . } or in ω \ {0} or in ω \ {0, 1}, and we construct such algebras. On the other hand, there exist algebras with two unary operations whose centralizer clones have essential arities that form an order filter of ω. We also construct (0, 1)-lattices for which S(cA) is a prescribed finite order ideal of ω \ {0} or of ω \ {0, 1}, and whose endomorphism monoid End01 A of all its (0, 1)-endomorphisms is isomorphic to a prescribed one. Finally, for a product K × L of (0, 1)-lattices (and other algebras), we give conditions under which the set S(c(K × L)) is the smallest possible, and show why these conditions are needed. In the process we prove that for any lattice A, the category \({\mathbb{L}}[A]\) of all lattices containing A and of their lattice homomorphisms is almost universal.

Derivation Tree Analysis for Accelerated Fixed-Point Computation

Abstract: We show that for several classes of idempotent semirings the least fixed-point of a polynomial system of equations is equal to the least fixed-point of a linearsystem obtained by "linearizing" the polynomials of in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was first used in [5] to show that Newton's method over commutative and idempotent semirings converges in a linear number of steps. Our results lead to efficient generic algorithms for computing the least fixed-point. We use these algorithms to derive several consequences, including an O(N3) algorithm for computing the throughput of a context-free grammar (obtained by speeding up the O(N4) algorithm of [2]), and a generalization of Courcelle's result stating that the downward-closed image of a context-free language is regular [3].

All idempotent hypersubstitutions of type (2,2)

Abstract: A hypersubstitution of type (2,2) is a map σ which takes the binary operation symbols f and g to binary terms σ(f) and σ(g). Any such σ can be inductively extended to a map \(\hat{\sigma}\) on the set of all terms of type (2,2). By using this extension on the set Hyp(2,2) of all hypersubstitutions of type (2,2) a binary operation can be defined. Together with the identity hypersubstitution mapping f to f(x1,x2) and g to g(x1,x2) the set Hyp(2,2) forms a monoid. This monoid is isomorphic to the endomorphism monoid of the clone of all binary terms of type (2,2). We determine all idempotent elements of this monoid. The results can be applied to the equational theory of Universal Algebra.

Telescope conjecture, idempotent ideals, and the transfinite radical

Abstract: We show that for an artin algebra $\Lambda$, the telescope conjecture for module categories is equivalent to certain idempotent ideals of mod-$\Lambda$ being generated by identity morphisms. As a consequence, we prove the conjecture for domestic standard selfinjective algebras and domestic special biserial algebras. We achieve this by showing that in any Krull-Schmidt category with local d.c.c. on ideals, any idempotent ideal is generated by identity maps and maps from the transfinite radical.

Associativity of the ∇ -operation on bands in rings

Abstract: Given a multiplicative band of idempotents S in a ring R, for all e,f∈S the ∇-product e ∇ f=e+f+fe−efe−fef is an idempotent that lies roughly above e and f in R just as ef and fe lie roughly below e and f. In this paper we study ∇-bands in rings, that is, bands in rings that are closed under ∇, giving various criteria for ∇ to be associative, thus making the band a skew lattice. We also consider when a given band S in R generates a ∇-band.

Combining Equational Tree Automata over AC and ACI Theories

Abstract: In this paper, we study combining equational tree automata in two different senses: (1) whether decidability results about equational tree automata over disjoint theories ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ imply similar decidability results in the combined theory${\mathcal{E}}_1 \cup {\mathcal{E}}_2$; (2) checking emptiness of a language obtained from the Boolean combinationof regular equational tree languages. We present a negative result for the first problem. Specifically, we show that the intersection-emptiness problem for tree automata over a theory containing at least one AC symbol, one ACI symbol, and 4 constants is undecidable despite being decidable if either the AC or ACI symbol is removed. Our result shows that decidability of intersection-emptiness is a non-modularproperty even for the union of disjoint theories. Our second contribution is to show a decidability result which implies the decidability of two open problems: (1) If idempotence is treated as a rule f(x,x) ?xrather than an equation f(x,x) = x, is it decidable whether an AC tree automata accepts an idempotent normal form? (2) If ${\mathcal{E}}$ contains a single ACI symbol and arbitrary free symbols, is emptiness decidable for a Boolean combination of regular ${\mathcal{E}}$-tree languages?

A cellular algebra with certain idempotent decomposition

Abstract: For a cellular algebra $\A$ with a cellular basis $\ZC$, we consider a decomposition of the unit element $1_\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\ZC$ can be partitioned into some pieces with good properties. Then by using a certain map $\a$, we give a coarse partition of $\ZC$ whose refinement is the original partition. We construct a Levi type subalgebra $\aA$ of $\A$ and its quotient algebra $\oA$, and also construct a parabolic type subalgebra $\tA$ of $\A$, which contains $\aA$ with respect to the map $\a$. Then, we study the relation of standard modules, simple modules and decomposition numbers among these algebras. Finally, we study the relationship of blocks among these algebras.

Extending modules relative to a torsion theory

Abstract: An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τI is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τI-extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τG we give an example of a module that is type 2 τG-extending but not extending.

Idempotent Boolean matrices and majorization

Abstract: We obtain a new structural characterization of idempotent Boolean matrices. This characterization allows us to describe all Boolean matrices that are majorized by a given idempotent.

The Maximum Idempotent Separating Congruence on E-inversive E-semigroups

Abstract: A semigroup S is an E-inversive E-semigroup if for every a ∈ S, there exists an element x ∈ S such that ax is idempotent and the set of all idempotents of S forms a subsemigroup. The aim of this paper is to investigate the maximum idempotent separating congruence on E-inversive E-semigroups by using a full and weakly self-conjugate subsemigroup.

On idempotent discrete uninorms

Abstract: This paper is devoted to classify all idempotent uninorms defined on the finite scale Ln = {0, 1, . . . , n}, called discrete idempotent uninorms. It is proved that any discrete idempotent uninorm with neutral element e ∈ Ln is uniquely determined by a decreasing function g : [0, e]→ [e, n] and vice versa. Based on this correspondence, the number of all possible discrete idempotent uninorms on a finite scale of n + 1 elements is given depending on n.

A note on idempotence-preserving maps

Abstract: Let Mn (F) be the space of all n × n matrices over a field F of characteristic not 2, and let Pn (F) be the subset of Mn (F) consisting of all n × n idempotent matrices. We denote by Φ n (F) the set of all maps from Mn (F) to itself satisfying A−λB∈Pn (F) implies for every A,B∈Mn (F) and λ∈F. In this note, we prove that φ∈Φ n (F) if and only if there exist δ∈{0,1} and an invertible matrix P∈Mn (F) such that either for every A∈Mn (F), or φ(A)=δPATP−1 for every A∈Mn (F). This improves the result of some related references.

Mutations of Absolute Valued Algebras

Abstract: Let A be an absolute valued algebra such that there exists a nonzero algebraic element e ∈ A satisfying some of the following conditions: 1. e(xy )= x(ey) for all x,y ∈ A. 2. (ex)e = e(xe) for all x ∈ A. We prove that the norm of A comes from an inner product. This generalizes previously known results in [21] and [10, 11] for the cases that e is a left unit and e is a central idempotent, respectively.

Regular elements and Green's relations in power Menger algebras of terms

Abstract: Dening an (n + 1)-ary superposition operation S n on the set W (Xn) of all n-ary terms of type , one obtains an algebra n clone := (W (Xn); S n , x1, . . . , xn) of type (n + 1, 0, . . . , 0). The algebra n clone is free in the variety of all Menger algebras ([9]). Using the operation S n there are dieren t possibilities to dene binary associative operations on the set W (Xn) and on the cartesian power W (Xn) n . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.

Homomorphisms and Congruences of Medial Semigroups with an Associate Subgroup

Abstract: Let S be the model of a semigroup with an associate subgroup whose identity is a medial idempotent constructed by Blyth and Martins considered as a unary semigroup. For another such semigroup T, we construct all unary homomorphisms of S into T in terms of their parameters. On S we construct all unary congruences again directly from its parameters. This construction leads to a characterization of congruences in terms of kernels and traces. We describe the K, T, L, U and V relations on the lattice of all unary congruences on S, again in terms of parameters of S.

Idempotent Mal'cev conditions and 2-uniform congruences

Abstract: A congruence is called 2-uniform if all of its blocks have exactly two elements. We prove that if A is a finite algebra that satisfies a nontrivial idempotent Mal’cev condition then any two 2-uniform congruences of A permute.