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Balanced d-lattices are complemented

100%

Goldstern M., Ploščica M.

Discussiones Mathematicae - General Algebra and Applications

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2002

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tom 22

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nr 1

33-37

EN

We characterize d-lattices as those bounded lattices in which every maximal filter/ideal is prime, and we show that a d-lattice is complemented iff it is balanced iff all prime filters/ideals are maximal.

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A partial order where all monotone maps are definable

100%

Goldstern M., Shelah S.

Fundamenta Mathematicae

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1997

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tom 152

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nr 3

255-265

EN

It is consistent that there is a partial order (P,≤) of size $ℵ_1$ such that every monotone function f:P → P is first order definable in (P,≤).

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Clones on regular cardinals

100%

Goldstern M., Shelah S.

Fundamenta Mathematicae

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2002

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tom 173

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nr 1

1-20

EN

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Discussiones Mathematicae - General Algebra and Applications

3 Goldstern M.

2 Shelah S.

1 Ploščica M.

2 2002

1 1997

Balanced d-lattices are complemented

A partial order where all monotone maps are definable

Clones on regular cardinals