On the functional completeness of simple tournaments

The theory of multitraces provides a new proof that any simple tournament with more than two elements is functionally complete.     

Remark on the completeness of an exponential type sequence

B. J. Birch [1] proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p ^a q ^b , where p, q are given coprime integers greater than 1. Subsequently, Davenport pointed out that the exponent b can be bounded in terms of p and q. N. Hegyvári [3] gave an effective version of this bound. In this paper, we improve this bound by reducing one step.     

The completeness of Peano multiplication

It is shown that the set of all theorems of Peano Arithmetic which mention only multiplication is a complete theory in the corresponding restricted language. The notion of a complete decidable covering of a theory is introduced. The author was partially supported by N.S.F. Grant No. MC578-02224.     

Problem of checking the functional completeness in the class of piecewise parallel functions

The author showed previously that it is possible to implement an arbitrary piecewise parallel function by circuits of linear elements and one nonlinear element implementing the θ-function. Thus, the functional completeness of a particular system of functions in the class of piecewise parallel functions was established. In this paper we consider the same class and study its completeness property for a system of functions whose closures contain all linear functions. We formulate and prove a criterion of functional completeness for such systems in terms of precomplete classes.     

The completeness of a polynomial vector field is determined by a transcendental trajectory

We prove that every polynomial vector field on C² that is complete on a transcendental (proper and non-algebraic) trajectory is complete in C².     

The completeness problem in partial hyperclones

The composition-closed sets of partial multi-valued operations, called partial hyperclones, defined on the finite set are investigated. It is shown that the lattice of all partial hyperclones is dually atomic, i.e., any non-full partial hyperclone is contained in a maximal partial hyperclone. Based on it some completeness criteria in the full partial hyperclone are established. Next the total list of maximal restriction-closed partial hyperclones is obtained and, thus, the completeness problem with respect to compositions and restrictions of partial hyperoperations is solved.