Well-ordered Set Research Articles

The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set.

The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set.

Asymptotic geometry of hyperbolic well-ordered cantor sets

In this paper we study the well-ordered Cantor sets in hyperbolic sets on the line and the plane. Examples of such sets occur in circle maps and in area-preserving twist maps. We set up a renormalization scheme employing in both cases the first return map. We prove convergence of this scheme. The convergence implies that the asymptotic geometry of such a well-ordered set with irrational rotation number and their nearby well-ordered orbits is determined by the Lyapunov exponent of this set.

Point processes indexed by directed sets

The concept of point process is defined where the parameter set is a directed set. By extending the parameter set, every point process can be described as an integer measure on a well-ordered set, and then the predictable projection of a point process can be constructed.

An accessibility proof of ordinal diagrams in intuitionistic theories for iterated inductive definitions

Let (/, -<) be a non-empty well-ordered system with the least element 0, and / be /W{co} with the largest element oo. Let A be a non-empty well-ordered set. Then 0(1, A) denotes the system of ordinal diagrams (o.d.'s) based on / and A. (cf. [9, §26].) The accessibility proof for 0(1, A) in [9, pp. 298-309] shows that every o.d. from 0(1, A) is accessible with respect to <t for every i in /. The central notions in this proof are /-fans and /-accessibility for i in /. Roughly speaking, an o.d. ft is an /-fan if for every /-</ and every /-section v of fi,v is /-accessible, and an o.d. is /-accessible if it is accessible in /-fans with respect to <*. Consider the case when the order type of (/, -<) is a successor ordinal £+1. If we formalize this accessibility proof for O(£+l, 1) (=0(1, 1)) naturally, then this proof can be done in the intuitionistic theory lD\+i for £+l-times iterated inductive definitions. The purpose of this paper is to show the following fact: the accessibility of each o.d. from 0(6+1, 1) with respect to <0 is derivable in DDL (Theorem)

Directed search through autobiographical memory.

Two experiments were designed to study retrieval from a well-ordered set of overlearned information in long-term autobiographical memory. In Experiment 1, each of 161 university students was asked to recall the name of one teacher from each of the 12 preuniversity school years. Recall was cued in forward (Grades 1-12), backward (Grades 12-1), or random order. Backward-ordered search proved most efficient: Fewer students failed to complete the task :in that search order, and those who did complete the task were faster than the successful students in the other search orders. Experiment 2 was similar to Experiment 1, except that the 148 students tested were asked to “think aloud” as they attempted recall. From the resulting protocols, 25 categories of verbal reports were identified, including four that reflected retrieval strategies. Omission errors (failures to recall teachers’ names) and, with one exception, frequencies of verbal reports decreased as recency increased. Taken together, the results of both experiments show that the recall of one item can indirectly aid the recall of contiguous items and that the probability of recalling an item from autobiographical memory is primarily a function of recency.

Homomorphisms and congruences on ωα-bisimple semigroups

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

Unique factorization monoids and domains

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M M is an ordered monoid and F F is a field, the ring F [ [ M ] ] F[[M]] of all formal power series with well-ordered support is a uf-domain iff M M is naturally ordered (i.e., whenever b &gt; a b &gt; a and a M ⋂ b M ≠ ∅ aM{ \bigcap ^b}M \ne \emptyset , then a M ⊂ b M ) aM \subset bM) .

On well-quasi-ordering transfinite sequences

Abstract. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i and j such that i &lt; j and qi ≤ qi. A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q. If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f(α) ≤ h(μ(α)) for every α ∈ A, we write f ≤ g. It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q. The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A(I) denote the set of ascending finite sequences of elements of a subset I of P. If s, t∈A(P), write s ≺ t if, for some m, the terms of s are the first m terms of t. Let T1,…,Tn be disjoint subsets of A(P) whose union T does not include two distinct sequences s, t such that s ≺ t. Then there exists an infinite subset I of P such that T ∩ A(I)is contained in a single Tj.

The Burali-Forti Paradox

The year 1897 saw the publication of the first of the modern logical paradoxes. It was published by Cesare Burali-Forti, the Italian mathematician whose name it has come to bear. Burali-Forti's own formulation of the paradox was not altogether satisfactory, as he had confused well-ordered sets as defined by Cantor with what he himself called “perfectly ordered sets” (“Classe parfettamente ordinata”). However, he soon realized his mistake, and published a note admitting the error and making the correction. He concluded the note with the observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the “perfectly ordered sets” for which it had first been obtained. We shall reproduce his results in their corrected form.