The algebra of implicit operations

For an ordinal and a class of topological algebras of a given type (which may be infinite and may contain inflnitary operations), an-aryimplicit operation on is any new-ary operation whose introduction does not eliminate any continuous homomorphisms between members of . The set of all-ary implicit operations on is denoted by and forms an algebra of the given type which is endowed with the least topology making continuous all homomorphisms into members of . With this topology, is a topological algebra in which the subalgebra of all-ary operations on which are induced by terms is dense, provided that is closed under the formation of closed subalgebras and finitary direct products. This is obtained by realizing as an inverse limit of-generated members of . These results are applied to pseudovarieties of topological and finite algebras.This work was supported, in part, by INIC grant 85/CEX/4. This paper was written while the author was a faculty member at the Universidade do Minho.Presented by Ralph McKenzie.     

Gorenstein curves and symmetry of the semigroup of values

LetO be the local ring of a irreducible algebroid curve and S its semigroup of values, Kunz in [7] proves thatO is a Gorenstein ring if and only if S is symmetrical. In this paper we give a generalization of this fact for the case of reduced curves with an arbitrary number of branches, d. For it we introduce a concept of symmetry for the semigroup of values S₊ ^d which generalizes the well known symmetry for d=1 (i.e. the irreducible case). This concept of symmetry is also closely related to the symmetry introduced by García in [4] (for the d=2 case) and the author in [3] (for arbitrary d) with the main goal of the explicit determination of S (in the case of plane curves).     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.