Showing papers on "Idempotence published in 1991"

When is a multiplicative derivation additive

Abstract: Our main objective in this note is to prove the following Suppose R is a ring having an idempotent element e ( e ≠ 0 , e ≠ 1 ) which satisfies: ( M 1 ) x R = 0 implies x = 0 ( M 2 ) e R x = 0 implies x = 0 ( and hence R x = 0 implies x = 0 ) ( M 3 ) e x e R ( 1 − e ) = 0 implies e x e = 0 If d is any multiplicative derivation of R , then d is additive

Semigroups with idempotent stabilizers and applications to automata theory

Abstract: We show that every finite semigroup is a quotient of a finite semigroup in which every right stabilizer satisfies the identities x = x^2 and xy = xyx. This result has several consequences. We first give a geometrical application : every finite transformation semigroup has a fixpoint-free covering (a transformation semigroup is fixpoint-free if every element which stabilizes a point is idempotent). Next we use our result and a result of I. Simon on congruences on paths to obtain a purely algebraic proof of a deep theorem of McNaughton on infinite words. Finally, we give an algebraic proof of a theorem of Brown on a finiteness condition for semigroups.

Pseudovarieties of completely regular semigroups

Abstract: We shall show that several results concerning the lattice of completely regular semigroup varieties find their analogues for the lattice of pseudovarieties of completely regular semigroups. We establish several complete idempotent endomorphisms and a subdirect decomposition of this lattice of pseudovarieties. These investigations culminate in Theorem 18 which is the analogue for pseudovarieties of Polak’s description [19] of the lattice of completely regular semigroup varieties. We shall in particular be able to describe the lattice of pseudovarieties of orthogroups in terms of the lattice of pseudovarieties of groups.

On the use of pairwise balanced designs and closure spaces in the construction of structures of degree at least 3

Abstract: We prove that a set of v-2 symmetric idempotent latin squares of order v , such that no two of them agree in a off-diagonal position, exists for all odd v>>0 . We describe how the techniques used in the proof relate to techniques used in [17] to construct generalised idempotent ternary quasigroups whose conjugate invariant group contains some specific subgroup. We also showhow these techniques fit into the more general context of trying to extend group divisible design methods to combinatorial structures, using closure spaces.

Idempotent Multiplications on Surfaces and Aspherical Spaces

Abstract: Every continuous idempotent multiplication on a space induces an idempotent comultiplication on its cohomology algebra over a commutative ring and a homomorphic idempotent multiplication on each homotopy group. We classify all idempotent comultiplications on any graded anticommutative algebra A∗ over a principal ideal domain K up to degree 2 provided the degree 1 component A1 is torsion free and the degree 2 component A2 is of rank 1. All algebraic possibilities can be topologically realized. We also describe all homomorphic idempotent multiplications on arbitrary groups. This allows a complete classification up to homotopy of all idempotent multiplications on aspherical CW-complexes. For surfaces we obtain an explicit list. Notably, the Klein bottle allows infinitely many nonhomotopic idempotent multiplications, but all other surfaces with nonabelian fundamental group have only the projections as idempotent multiplications (up to homotopy). Introduction. Idempotent multiplications on sets and topological spaces have been considered by many authors, for instance as an axiomatic approach to the averaging operation (sample: [2, 3, 10]). If X denotes a connected topological space, then the existence of H-space structures places severe restrictions on the structure of X. (See, for instance, [6] or [16].) This is due to the presence of homotopy identities on both sides. If, however, one considers idempotent multiplications μ : X ×X → X, that is, multiplications which satisfy μ(x, x) = x for all x ∈ X, then no restriction follows from the presence of such multiplications, since every space X allows the two idempotent multiplications p1, p2 : X × X → X, p(x, y) = x and q(x, y) = y for all x, y ∈ X. These are the so-called trivial multiplications. On the other hand, the existence of nontrivial idempotent multiplication again forces restrictions on the space. We wish to illustrate this by discussing idempotent multiplications on suitable classes of spaces. The Received by the editors on September 1, 1988, and in revised form on October 24, 1988. Copyright c ©1991 Rocky Mountain Mathematics Consortium

Idempotent algebras with log-linear free spectra

Abstract: In this paper we consider idempotent algebras whosepn-sequences (the sequences of the numberspn of essentiallyn-ary polynomials) have a subexponential rate of growth. Studying the symmetry groups of polynomials we establish some consequences of this property. In particular, a new characterization of semilattices is obtained, and all idempotent commutative algebras with log-linear free spectra are described.

Extraction of functional from logic program

Abstract: This paper shows a method of extracting a functional from a logic program, by means of a dataflow dealing with sequences from the set of idempotent substitutions. The dataflow is expressed as a functional involving fair merge functions in order to represent the atom set union over a sequence domain, as well as functions to act on unifiers, to reflect the unit resolution deductions virtually. The functional completely and soundly denotes the atom generation in terms of idempotent substitutions without using atom forms. Its least fixpoint is interpreted as denoting the whole atom generation in terms of manipulations on idempotent substitutions.

uivalence for empotent ri

Abstract: Garcia, J.L. and J.J. Sim6n. Morita equivalence for idempotent rings. Journal of Pure and Applied Algebra 76 (1991) 39-56. In this paper, the classical theory of Morita equivalence is extended to idempotent rings which do not necessarily have an identity element. 111 this case. the role of progenerators is played by the unital and codivisible modules w: ; I generate :II the unital modules.

ldempotents in Nest Algebras

Abstract: The algebraic equivalence and similarity classes of idempotents within a nest algebra Alg/l are completely characterized. We obtain necessary and sufficient conditions for two idempotents to he equivalent or similar. Our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a “dimension function” from pxb into Nv {IX }. We complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents. Also, to each sequence of dimension functions, there corresponds a commuting sequence of idempotents. A criterion is obtained for when an idempotent is similar to a subidempotent of another. The mapping which sends an equivalence class (or idempotent) to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a Type I W*-algebra to its center valued trace. WC prove that almost commuting, similar idempotents are homotopic; this contrasts with the situation in certain C*-algebras. Using this, we show that similar, stmultaneously diagonalizable idempotents are homotopic, and in the continuous nest case, every diagonal idempotent is homotopic to a core projection.

A class of inverse semigroups with boolean congruence lattices

Abstract: A construction of inyerse semigroups whose idempotents form a (locally finite) tree and whose congruence lattices have the property P is given where P stands for one of the fol- lowing properties of lattices: (dually) sectionally complemented, relatively complemented, modular and complemented, Boolean, respectively. These semigroups are completely character- ized up to: congruence-free inverse semigroups (without zero), simple groups and locally finite trees. Furthermore, special sublattices of the congruence lattice easily can be studied: any two trace classes are isomorphic, and the lattices of all semilattice congruences and idempotent pure congruences, respectively are Boolean.