Akihiro Nozaki

The number of orthogonal permutations

A problem on maximal clones in universal algebra leads to the natural concept of orthogonal orders and their characterization. Two (partial) orders on the same set P are orthogonal if they share only trivial endomorphisms, i.e. if the identity self-map of P is the sole non-constant self-map preserving (i.e. compatible with) both orders. We start with a neat and easy characterization of orthogonal pairs of chains (i.e. linear or total orders) and then proceed to the study of the number q(k) of chains on {0, 1, …, k − 1} orthogonal to the natural chain 0 < 1 < ⋯ < k − 1. We obtain a recurrence formula for q(k) and prove that the ratio q(k)k! (of such chains among all chains) goes to e−2 = 0.1353 ⋯ as k → ∞. Results are formulated in terms of permutations.

Semirigid sets of central relations over a finite domain

A set of central h-ary relations on a set A is called semirigid if the clones of k-valued logic functions determined by the relations share only the clone K/sub h-1/ consisting of all projections and all functions assuming at most h-1 values (12; K/sub 1/ is the set of trivial functions, i.e., the clone consisting of all constants and all projections). The problem of determining semirigid sets of central relations is studied. For the set of h-ary central relations with the centers of the largest size, it is shown that the set consisting of all such relations is the only semirigid set. It is also shown that the minimum size of a semirigid set of central h-ary relations is h+1. For k=4, semirigid sets of binary central relations are investigated in detail.<<ETX>>

Hereditary clones of multiple valued logic algebra

We discuss relationships between properties and operations over the set /spl Omega/ of MVL functions. Closed properties are those invariant under the classical closure operation. A new type of properties, called hereditary, is defined, as well as hereditary closure. We calculate the ratio of hereditary properties, describe the families of maximal hereditary clones, and give a formula for their enumeration. We show that there are exactly eleven such clones in ternary logic. For Boolean algebra the lattice of all hereditary classes is finite, and we describe it completely. Meanwhile, starting from the three valued case there are still a continuum number of clones.<<ETX>>

Semirigid sets of quasilinear clones

Let k be a prime and G a Galois field on k:=(0,1,. . .,k-1). The set of all quasilinear (or affine with respect to G) k-valued logic functions is a maximal clone called quasilinear. A family of quasilinear clones on k is semirigid if the clones of the family share exactly the constant functions and the projections. Semirigid sets of quasilinear clones are needed for the classification of bases of k-valued logic, which is unknown for k>3. The authors characterize all semirigid sets of quasilinear clones. In particular, for k=5 they describe all semirigid triples of quasilinear clones and show that no such pair exists. For every prime k>5 they exhibit a semirigid pair of quasi-linear clones. The techniques used are based on elementary number theory and on polynomials over G.<<ETX>>

Sorting using networks of deques

The number of required deques for sorting all sequences of n items in a parallel or series network of deques is considered. It is shown that the optimal number of required deques is O(n12) for a parallel network, while it is O(log n) for a series network. These orders, O(n12) and O(log n), also remain valid for the networks of restricted deques.