Showing papers in "Archiv der Mathematik in 1973"

Valuations on Distributive Lattices I

Abstract: In Parts I and II [Arch. Math. 24, 230–239, 337–345 (1973)] we were principally interested in combinatorial applications of the valuation ring of a distributive lattice. We now show how this ring provides a natural setting for some elementary results in measure theory as well as some classical results on representations of distributive lattices. Specifically, in the valuation ring V(L) of a distributive lattice L it is easy to identify various extensions of L as well as prime ideals of L and so arrive at some theorems of Pettis, Birkhoff, and Stone. For any faithful representation of L as a lattice of sets the extension of V (L) by real (or complex) scalars is naturally isomorphic to the algebra of simple functions, and the sup norm on the functions comes from an intrinsic norm on VR(L). The Stone space of L corresponds to the spectrum of VR(L) with the Zariski topology.

Elementary geometric applications of a maximum principle for nonlinear elliptic operators

Abstract: The max imum principle for nonlinear, second-order, elliptic operators has been used in Differential Geometry to obtain various uniqueness theorems, characterizations of the sphere and results on Weingarten surfaces [1, 4]. In this note we shall show how it can be applied to yield information on the size of immersed, compact surfaces. A weak form of this maximum principle will be sufficient for our purposes, which we now formulate. Let u (x, y), v (x, y) be real-valued/unctions o/clas~ C z in some bounded domain .(2, satis/ying in Q the inequality F (u~, u~, u= , u~v, uvv) >= F(v~, vv, v= , vxy, Vyv), where F ks real-valued o /c lass C 1 /or all values o / i t s arguments. Assume that u ~_ v in .(2 and that F is positive de/inite elliptic with respect to the/unctions z = O u + ( 1 O ) v , 0 < 0 _ < 1 ,

Monads and monoids on symmetric monoidal closed categories

Abstract: part of a $/%monad A. In this note, we show that for a monad Tand a monoid A and a Y/'-natural transformation r -- (~ A --> T the following are equivalent (1) q is a Yf-monad map; (2) q is monoidal W-natural transformation; (3) the map Oq: A -~ T I corresponding to q under the bijection mentioned above is a monoid map. As a consequence we show that for every $P-monad T on Y/\" there is a monoid (namely TI) and a Y/~-monad map z: TI---> T such that if A is a monoid and ~.: A -~ Tis a Y/--monad map then there exists a unique $/~-monad map fl: A --> TI with T �9 fl = a. We also show that if T is a commutative monad (i.e. the two canonical monoidal structures on T agree [3], p. 7) then TI is a commutative monoid and thus TI is a commutative monad. We also note that there are non-commutative monads T such that TI is a commutative monoid.