Showing papers on "Idempotence published in 1982"
01 Jan 1982
TL;DR: For the most part in the lectures the authors shall concentrate on the finitary domains, but the continuous domains find an interest as a generalization of interval analysis and by the connection with spaces of upper-semicontinuous functions.
Abstract: Strictly speaking the structures to be used are not lattices since as posets they will lack the top (or unit) element, but the adjunction of a top will make them complete lattices. The closure properties as posets, then, are closure underinf of any non-empty subset and sup of directed subsets. A family of subsets of a set closed underintersections of non-empty subfamilies and unions of directed subfamilies is a special type of poset with the closure properties where additionally every element is the directed sup (union) of the finite (“compact”) elements it contains. We call such posets finitary domains. (With a top they are just the well known algebraic lattices.) The continuous domains can be defined as the continuous retracts of finitary domains. A mapping between domains is continuous if it preserves direct sups. A map of a domain into itself is aretraction if it is idempotent. Starting with a finitary domain, the range (= fixed-point set) of a continuous retraction — as a poset — is a continuous domain. Numberless characterizations of continuous domains, both topological and order-theoretic, can be found in [2]. For the most part in the lectures we shall concentrate on the finitary domains, but the continuous domains find an interest as a generalization of interval analysis and by the connection with spaces of upper-semicontinuous functions.
60 citations
TL;DR: In this paper, the authors proposed a semigroup with the additional relation of an additional relation between the semigroup's members and the associated relations, which is called aband or anidempotent semigroup.
Abstract: Let B be a semigroup with the additional relation
$$\begin{gathered} xx \Rightarrow x \hfill \\ xyz \Rightarrow xz if x \mathop {CI}\limits_ = z and xy\mathop {CI}\limits_ = z \hfill \\ \end{gathered} $$
B is called aband or anidempotent semigroup [3].
34 citations
TL;DR: In this paper, it was shown that if a cell (i,i) is occupied by a symbol, for each i, 1 < i < n, then the partial latin square is idempotent.
Abstract: A partial latin square on t symbols
24 citations
01 Sep 1982
TL;DR: In this paper, the rank additivity condition of matrices and polynomial equations satisfied by these matrices was studied. But the relation between rank additive conditions and the rank-additivity condition was not discussed.
Abstract: : Various results are given concerning the logical relation between the rank additivity condition of matrices and polynomial equations satisfied by these matrices, generalizing earlier results on idempotent, tripotent, and r- potent matrices.
18 citations
TL;DR: In this article, the concept of a connection between partially ordered sets was introduced, which is a natural generalization of the Galois connection, and retraction operators (idempotent maps) played the role of closure operators.
15 citations
10 citations
TL;DR: The trace of the lifting of an idempotent is independent of lifting and is a function only of the equivalence class of a nonzero idemomorphism as discussed by the authors, and it is always a p-adic integer.
9 citations
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TL;DR: In this paper, the authors partially solve the problem posed by E. Cech by showing that there is a nontrivial closure operator which is onto, that is, for which in some sense every subset of our space is closed.
4 citations
2 citations
TL;DR: In this paper, the authors define the notion of nilpotency of elements in a polynomial algebra over a field of characteristic not 2 or 3 and prove that the algebra of polynomials in y over y is associative.