Hajime Machida

Gigantic pairs of minimal clones

L. Szabo (1992) asked for the minimal number n=n(|A|) such that the clone of all operations on A can be generated as the join of n minimal clones. He showed, e.g., n(p)=2 for any prime p, and later G. Czedli (1998) proved that if k has a divisor /spl ges/5 then n(k)=2. In this paper, a pair (f, g) of operations is called gigantic if each of f and g generates a minimal clone and the set {f, g} generates the clone of all operations. First, we give a general theorem to characterize a gigantic pair. Then we show that n(k)=2 for every k which is not a power of 2.

Relations between clones and full monoids

An endoprimal clone is defined via a set of unary operations. It was known before that the endoprimal clone for the set O/sub 4//sup (1)/ of all unary operations on, a k-element set is the least clone J/sub k/ and that the endoprimal clone for the symmetric group S/sub k/ strictly includes J/sub k/. In this paper we consider monoids of unary operations and clones corresponding to such monoids. We define a descending sequence {N/sub i/}/sub i=1//sup k=1/ of monoids lying between O/sub k//sup (1)/ and S/sub k/, and show that the endoprimal clone for N/sub k-1/ is distinct from J/sub k/. Finally we present a characterization of the endoprimal clone for S/sub k/.

On the intersection of maximal partial clones and the join of minimal partial clones

Let A be a nonsingleton finite set and M be a family of maximal partial clones with trivial intersection over A. What is the smallest possible cardinality of M? Dually, if F is a family of minimal partial clones whose join is the set of all partial functions on A, then what is the smallest possible cardinality of F? The purpose of this note is to present results related to these two problems.

Some continuous maps on the space of clones in multiple-valued logic

The lattice L/sub k/ of all clones over the set {0, 1,/spl middot//spl middot//spl middot/, k-1} is known to be a metric space. In this paper, we define some maps induced by the lattice operators and note that those induced by the meet operator are continuous maps from L/sub k/ to L/sub k/. Secondly, we use the meet operator to construct two continuous maps from L/sub 3/ to L/sub 2/. These maps are shown to be order-preserving and surjective. Finally, the images of all the maximal clones in L/sub 3/ and those of Yanov-Muchnik clones in L/sub 3/ under these maps are studied.