BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-72-269
       ENTRY:: October 16, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Arithmetic properties of certain recursively defined sets.
        TYPE:: Technical Report
      AUTHOR:: Klarner, David A.
      AUTHOR:: Rado, Richard
        DATE:: March 1972
       PAGES:: 31
    ABSTRACT:: Let R denote a set of linear operations defined on the set P
               of positive integers; for example, a typical element of R has
               the form $\rho (x_1, \ldots ,x_r) = m_0 + m_1 x_1 + \ldots +
               m_r x_r where m_0, \ldots ,m_r$ denote certain integers.
               Given a set A of positive integers, there is a smallest set
               of positive integers denoted <R:A> which contains A as a
               subset and is closed under every operation in R. The set
               <R:A> can be constructed recursively as follows: Let $A_0$ =
               A, and define

               $A_{k+1} = A_k \cup \{\rho (\bar{a}): \rho \in R,\bar{a}\in
               A_k \times \ldots \times A_k\}$ (k = 0,1,\ldots ),

               then it can be shown that <R:A> = $A_0 \cup A_1 \cup \ldots
               $. The sets <R:A> sometimes have an elegant form, for
               example, the set <2x + 3y: 1> consists of all positive
               numbers congruent to 1 or 5 modulo 12. The objective is to
               give an arithmetic characterization of elements of a set
               <R:A>, and this paper is a report on progress made on this
               problem last year. Many of the questions left open here have
               since been resolved by one of us (Klarner).
       NOTES:: [Adminitrivia V1/Prg/19951016]
         END:: STAN//CS-TR-72-269