The typical countable algebra

We argue that it makes sense to talk about ``typical'' properties of lattices, and then show that there is, up to isomorphism, a unique countable lattice L* (the Fraisse limit of the class of finite lattices) that has all ``typical'' properties. Among these properties are: L* is simple and locally finite, every order preserving function can be interpolated by a lattice polynomial, and every finite lattice or countable locally finite lattice embeds into L*. The same arguments apply to other classes of algebras assuming they have a Fraisse limit and satisfy the finite embeddability property

An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

If (B_x: x in N) is a Borel family of sets, indexed by the Baire space N = omega^omega, all B_x have measure zero, and the family is increasing, then the union of all B_x also has measure zero. We give two proofs of this theorem: one in the language of set theory, using Shoenfield's theorem on Sigma-1-2 sets, the other in the language of probability theory, using von Neumann's selection theorem, and we apply the theorem to a question on completely uniformly distributed sequences

Interpolation in ortholattices

If L is a complete ortholattice, f any partial function from L^n to L, then there is a complete ortholattice L* containing L as a subortholattice, and an ortholattice polynomial with coefficients in L* which represents f on L^n. Iterating this construction long enough yields a complete ortholattice in which every function can be interpolated by a polynomial on any set of small enough cardinality.Comment: 6 pages. See also http://info.tuwien.ac.at/goldstern/papers/index.html#orth

Analytic clones

We use a method from descriptive set theory to investigate the two precomplete clones above the unary clone on a countable set

Completion of Semirings

A semiring can be ``completed'' (i.e., embedded into a semiring in which all infinite sums are defined and satisfy some reasonable properties) iff this semiring can be naturally partially ordered. This construction is ``natural'' (a left adjoint to the forgetful functor), and quite straightforward.Comment: This is an English summary of my 1985 (unpublished) diploma thesi

Lattices, interpolation, and set theory

We review a few results concerning interpolation of monotone functions on infinite lattices, emphasizing the role of set-theoretic considerations. We also discuss a few open problems

The Complexity of Fuzzy Logic

Lukasiewicz logic is a "fuzzy" logic in which truth value can be real numbers in the unit interval. There are connectives for min, max, addition and complement (1-x). The "value" of a closed formula in a fuzzy (relational model) is defined in the natural way. A formula is called valid iff it has value 1 in every fuzzy model. We show that the set of valid formulas in Lukasiewicz predicate logic is a complete Pi^0_2 set. We also show that if we restrict our attention to the classical language (min, max, complement) then the classically valid formulas are exactly those formulas whose fuzzy value is 1/2

A Note on Superamorphous Sets and Dual Dedekind-Infinity

We give a simple example of a set that is weakly Dedekind infinite (= can be mapped onto omega) but dually Dedekind finite (=cannot be mapped noninjectively onto itself), namely, the power set of a superamorphous set. (A infinite set is superamorphous if all finitary relations on it are definable in the language of equality.) We also show that the property of "inexhaustibility" is not closed under supersets unless the full axiom of choice holds

Order polynomially complete lattices must be LARGE

If L is an order polynomially complete lattice, (that is: every monotone function from L^n to L is induced by a lattice-theoretic polynomial) then the cardinality of L is a strongly inaccessible cardinal. In particular, the existence of such lattices is not provable in ZFC, nor from ZFC+GCH. Although the problem originates in algebra, the proof is purely set-theoretical. The main tools are partition and canonisation theorems. It is still open if the existence of infinite o.p.c. lattices can be refuted in ZFC.Comment: This is paper number GoSh:633 in Shelah's list. The paper is to appear in Algebra Universalis

Clones on regular cardinals

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^kappa many maximal (=precomplete) clones on a set of size kappa." The clones we construct here do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem we show that for many cardinals kappa there are 2^2^kappa many such clones on a set of size kappa. Finally, we show that on a weakly compact cardinal there are exactly 2 maximal clones which contain all unary functions