Monomorphism

Abstract: Let R be a finite dimensional hereditary algebra. We are concerned with the problem of determining the indecomposable R-modules of finite length. This problem completely has been solved in the case when R is of finite or of tame representation type, but seems to be rather hopeless in the case of wild representation type. In this situation, the only known classes of modules are the socalled pre-projective and the pre-injective ones. The remaining indecomposable modules are called regular. In this paper, we want to initiate the study of the regular modules. The result we obtain seems to be rather surprising: we will show that the regular modules behave rather similar to modules over a serial algebra. In order to state the main theorem, we need the notion of an irreducible homomorphism, which was introduced by Auslander and Reiten [-3]. Let X and Y be two non-zero R-modules. A homomorphism f : X ~ Y is said to be irreducible, if it is neither a split monomorphism, nor a split epimorphism, and if for any factorisation X ~ I ~ Y o f f , either f is a split monomorphism o r f is a split epimorphism. Note that an irreducible homomorphism is always either a monomorphism or an epimorphism. A non-zero R-module S will be called quasisimple, ifS is regular, and there is no irreducible monomorphism of the form U ~ S with U non-zero. In this case, we will call the map 0 ~ S irreducible.

Abstract: ABsmAcT.-We present a survey of passerine birds designed to investigate the frequency with which sexual dimorphism in coloration or color pattern has evolved from monomorphism (or the converse). Based on the number of genera that have both a monomorphic and a dimorphic species, and the minimum number of changes inferred to have occurred between genera, the transition between dimorphism and monomorphism has occurred at least 150 times. Using the Sibley/Ahlquist phylogeny, we obtain maximum likelihood estimates of the probability that one state will be in the other after one million years of 0.01 to 0.02 (monomorphism to dimorphism) and 0.03 to 0.04 (dimorphism to monomorphism). The rate of transition from dimorphism to monomorphism appears to be higher than the converse, and there are more monomorphic than dimorphic species. We conclude that the transition between alternative states is not difficult, and that the evolution of sexual dimorphism, given appropriate selection pressures, is unlikely to be constrained. Received 15 August 1995, accepted 1 April 1996.