Showing papers on "Idempotence published in 1990"
TL;DR: In this paper, it was shown that the idempotent rank of K(n, r) is the Stirling number of the second kind, defined as the cardinality of a minimal generating set of idempots.
Abstract: The subsemigroup Sing n of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r) , the Stirling number of the second kind.
115 citations
TL;DR: In this paper, it was shown that every finite-dimensional absolute valued algebra is an inner product space (Theorem 3.6) and that every algebra with a central idempotent is an algebra with an involution.
36 citations
TL;DR: In this article, it was shown that any complex square matrix T is a sum of finitely many idempotent matrices if and only if trT is an integer and trT ⩾ rank T. The problem of the minimum number of idempots needed to sum T and obtain some partial results.
36 citations
TL;DR: The existence of idempotent elements, that is elements e, e # 0, such that e = e, is of interest in the study of a non-associative algebra.
Abstract: A finite-dimensional commutative algebra A over a field K is called a Bernstein algebra if there exists a non-trivial homomorphism co: A -> K (baric algebra) such that the identity (x) = CO(X)JC holds in A (see [7]). The origin of Bernstein algebras lies in genetics (see [2,8]). Holgate (in [2]) was the first to translate the problem into the language of non-associative algebras. Information about algebraic properties of Bernstein algebras, as well as their possible genetic interpretations, can be found in [10, Chapter 9B; 11; 12; 3; 1]. The existence of idempotent elements, that is elements e, e # 0, such that e = e, is of interest in the study of the structure of a non-associative algebra. From the biological aspect the existence of such elements is also interesting, because the equilibria of a population which can be described by an algebra correspond to idempotent elements of this algebra. The algebras occurring in applications usually do contain an idempotent. This occurs in Bernstein algebras (see [10]). With respect to an idempotent eeA (whose existence is guaranteed), A splits into the direct sum A = (e) + U+Z, where
24 citations
TL;DR: In this article, the lattice theoretic properties of these mappings and the interactions between them have been studied in the context of completely regular semigroups (CR) with the associative binary operation of multiplication and the unary operation of inversion subject to the laws x = xx-1, (x−1)-1 = x and xx − 1 = x-1x) is studied.
Abstract: The class CR of completely regular semigroups (unions of groups or algebras with the associative binary operation of multiplication and the unary operation of inversion subject to the laws x = xx-1, (x−1)-1 = x and xx-1 = x-1x) is a variety. Among the important subclasses of CR are the classes M of monoids and I of idempotent generated members. For each C ∈ {I, M}, there are associated mappings ν → ν ∩ C and ν → (Ν ∩ C), the variety generated by ν ∩ C. The lattice theoretic properties of these mappings and the interactions between these mappings are studied.
22 citations
TL;DR: Theorem 10 below is a generalization of the result of Evans and Ganter as discussed by the authors, which states that every algebra has a modular subalgebra lattice for groups of quite different structure (see Iwasawa [2] ).
Abstract: Varieties in which every algebra has a modular subalgebra lattice were studied by Trevor Evans and Bernhard Ganter [1]. They proved that every such variety is Hamiltonian (i.e., any subalgebra is a congruence class of a suitable congruence), and if, in addition, the variety is idempotent (i.e., every one element subset is a subalgebra), then it is a variety of trivial algebras in the sense that every operation is a projection. Moreover, they gave a characterization of the subalgebra-rnodular varieties through a condition for all ternary terms of the variety. We will restate this result as Theorem 10 below. Our experience with groups taught us that while the subgroup lattice is modular for groups of quite different structure (see Iwasawa [2]), the modularity of the subgroup lattice of the direct square G x G implies that G is Abelian (Lukfics and Pfilfy [41). This simple observation inspired the general question, what can be deduced if the modularity of the subalgebra lattice Sub (A x A) is assumed. It turned out that the results of Evans and Ganter can be generalized. Namely, it follows that A is Hamiltonian and whenever A is idempotent it must be trivial, with the exception of one two-element algebra. In order to prove that the algebra is Abelian (in the sense that it satisfies the Term Condition (TC)), we had to assume that Sub (A 4) is modular. Furthermore, the distributivity of Sub (A 3) implies that the algebra is strongly Abelian, i.e., it satisfies the strong Term Condition (TC*) introduced by McKenzie [5]. We consider modularity and distributivity of the subalgebra lattice for direct products, because these properties are inherited by factor algebras and also, trivially, by subalgebras. However, we give a simple example where the subalgebra lattice of some factor algebra A/O does not belong to the lattice variety generated by Sub A.
10 citations
TL;DR: The n-th member of the growth sequence of a globally idempotent finite semigroup without identity element is at least 2n as discussed by the authors, which was conjectured by Wiegold.
Abstract: The n–th member of the growth sequence of a globally idempotent finite semigroup without identity element is at least 2n. (This had been conjectured by J. Wiegold.)
6 citations
01 Apr 1990
TL;DR: In this article, a differentiable semigroup is defined as a topological semigroup in which the associative multiplication function is continuously differentiable and there is an open set U containing e so that there is a C retraction 1 of U into the set of idempotents of S so that x and y are in the maximal subgroup of S determined by D(x) for each x in U. This leads to a natural decomposition of S near e into a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections
Abstract: A differentiable semigroup is a topological semigroup (S, *) in which S is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If e is an idempotent element of such a semigroup we show that there is an open set U containing e so that there is a C retraction 1 of U into the set of idempotents of S so that ?(x)?D(y) = ?I(x) for x and y in U and x?(x) is in the maximal subgroup of S determined by D(x) for each x in U. This leads to a natural decomposition of S near e into the union of a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections with U are the point inverses under (. In case S is the semigroup under composition of continuous linear transformations on a Banach space, in the case of a nontrivial idempotent e, the existence of (D implies that operators near an e have nontrivial invariant subspaces. A dual right handed result holds.
1 citations
01 Jan 1990
TL;DR: In this article, the authors adopt another approach to the concept of the kernel of an idempotent separating congruence on a regular semigroup and give a survey of some of the results obtained in collaboration with K. S. Nambooripad.
Abstract: Let S be a regular semigroup and E(S) its set of idempotents. Let θ be an idempotent separating congruence on S. Traditionally by the kernel ker θ of θ we understand the union of the idempotent θ-classes (see e.g. [4]). For an idempotent separating congruence θ the idempotent θ-classes are groups, namely the θ-classes containing idempotents. The kernel normal system of θ considered by Preston in [6] is the set of these idempotent θ-classes and contains more information than the above mentioned ker θ, which, after all, is just a subset of S. In the following we shall adopt still another approach to the concept of the kernel of an idempotent separating congruence on a regular semigroup. We shall give a survey of some of the results obtained in collaboration with K. S. S. Nambooripad.
1 citations