ACTA issues

The number of unary clones containing the permutations on an infinite set

Michael Pinsker

Acta Sci. Math. (Szeged) 71:3-4(2005), 461-467

Abstract. We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, on some cardinals as large as the whole clone lattice. Moreover we find that, with one exception, even the cardinalities of the intervals between the monoid of all permutations and the maximal submonoids of the full transformation monoid are as large. Whether or not the only exception is of the same cardinality as the other intervals depends on additional axioms of set theory.

AMS Subject Classification (1991): 08A40, 08A05

Keyword(s): clone lattice, permutations, unary clones, transformation monoid, submonoids

Received October 20, 2004, and in revised form May 20, 2005. (Registered under 5882/2009.)