Lattices with large minimal extensions

This paper characterizes those finite lattices which are a maximal sublattice of an infinite lattice. There are 145 minimal lattices with this property, and a finite lattice has an infinite minimal extension if and only if it contains one of these 145 as a sublattice. Received October 5, 1998; accepted in final form May 19, 1999.     

Uniqueness of the Riemann minimal examples

We prove that a properly embedded minimal surface in R ³ of genus zero with infinite symmetry group is a plane, a catenoid, a helicoid or a Riemann minimal example. We introduce the language of Hurwitz schemes to understand the underlying moduli space of surfaces in our setting. Oblatum 30-V-1997 & 5-VIII-1997     

Uniqueness of form extensions and domination of semigroups

In this article, we study uniqueness of form extensions in a rather general setting. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. Our main abstract result transfers uniqueness of form extension of a dominating form to that of a dominated form. This result can be applied to a multitude of examples including various magnetic Schrödinger forms on graphs and on manifolds.     

Joint consistency in extensions of the minimal logic

Analogs of Robinson’s theorem on joint consistency are found which are equivalent to the weak interpolation property (WIP) in extensions of Johansson’s minimal logic J. Although all propositional superintuitionistic logics possess this property, there are J-logics without WIP. It is proved that the problem of the validity of WIP in J-logics can be reduced to the same problem over the logic Gl obtained from J by adding the tertium non datur. Some algebraic criteria for validity of WIP over J and Gl are found.     

N-free orders and minimal interval extensions

We investigate problems related to the set of minimal interval extensions of N-free orders. This leads us to a correspondence between this set for an arbitrary order and a certain set of its maximal N-free reductions. We also get a 1-1-correspondence between the set of linear extensions of an arbitrary order and the set of minimal interval extensions of the linegraph of that order. This has an algorithmic consequence, namely the problem of counting minimal interval extensions of an N-free order is #P-complete. Finally a characterization of all N-free orders with isomorphic root graph is given in terms of their lattice of maximal antichains; the lattices are isomorphic iff the root graphs agree.This work was supported by the PROCOPE Program. The first author is supported by the DFG.     

Measure preserving extensions and minimal wandering rates

For conservative ergodic (infinite) measure preserving transformations we show that existence and asymptotics of minimal wandering rates are preserved if we pass to extensions or factors. Consequently, they are similarity invariants.     

Parametrization of minimal rank block Hankel matrix extensions and minimal partial realizations

A parametrization of allm-rowsn-columns extensions of block Hankel matrices with minimal rank is presented. As a particular case, a new parametrization of the minimal partial realizations of a sequence of blocks is given. In this way results of O.H. Bosgra [2], I. Gohberg/M.A. Kaashoek/L. Lerer [3], and H. Woerdeman [13] are extended.     

Uniqueness of minimal coverings of maximal partial clones

A partial function f on a k-element set E _k is a partial Sheffer function if every partial function on E _k is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone on E _k , a characterization of partial Sheffer functions reduces to finding families of minimal coverings of maximal partial clones on E _k . We show that for each k ≥ 2, there exists a unique minimal covering.     

The lattice of extensions of the minimal logic

In this article, we survey the results on the lattice of extensions of the minimal logic Lj, a paraconsistent analog of the intuitionistic logic Li. Unlike the well-studied classes of explosive logics, the class of extensions of the minimal logic has an interesting global structure. This class decomposes into the disjoint union of the class Int of intermediate logics, the class Neg of negative logics with a degenerate negation, and the class Par of properly paraconsistent extensions of the minimal logic. The classes Int and Neg are well studied, whereas the study of Par can be reduced to some extent to the classes Int and Neg.