Clones on regular cardinals

Martin Goldstern, and Saharon Shelah. "Clones on regular cardinals." Fundamenta Mathematicae 173.1 (2002): 1-20. <http://eudml.org/doc/282660>.

@article{MartinGoldstern2002,
abstract = {We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are $2^\{2^λ\}$ maximal (= “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem from pcf theory we show that for cardinals λ (in particular, for all successors of regulars) there are $2^\{2^λ\}$ such clones on a set of size λ. Finally, we show that on a weakly compact cardinal there are exactly 2 precomplete clones which contain all unary functions.},
author = {Martin Goldstern, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {precomplete clones; maximal clones; weakly compact cardinal; negative square bracket partition relation; regular cardinals; Jonsson algebras; lattice of clones; ultrafilters; maximal closed classes of operations},
language = {eng},
number = {1},
pages = {1-20},
title = {Clones on regular cardinals},
url = {http://eudml.org/doc/282660},
volume = {173},
year = {2002},
}

TY - JOUR
AU - Martin Goldstern
AU - Saharon Shelah
TI - Clones on regular cardinals
JO - Fundamenta Mathematicae
PY - 2002
VL - 173
IS - 1
SP - 1
EP - 20
AB - We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are $2^{2^λ}$ maximal (= “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem from pcf theory we show that for cardinals λ (in particular, for all successors of regulars) there are $2^{2^λ}$ such clones on a set of size λ. Finally, we show that on a weakly compact cardinal there are exactly 2 precomplete clones which contain all unary functions.
LA - eng
KW - precomplete clones; maximal clones; weakly compact cardinal; negative square bracket partition relation; regular cardinals; Jonsson algebras; lattice of clones; ultrafilters; maximal closed classes of operations
UR - http://eudml.org/doc/282660
ER -