Green's relations, regularity and abundancy for semigroups of quasi-onto transformations

Let V be an infinite-dimensional vector space and for every infinite cardinal n such that n≤dimV, let AE(V,n) denote the semigroup of all linear transformations of V whose defect is less than n. In 2009, Mendes-Gonçalves and Sullivan studied the ideal structure of AE(V,n). Here, we consider a similarly-defined semigroup AE(X,q) of transformations defined on an infinite set X. Quite surprisingly, the results obtained for sets differ substantially from the results obtained in the linear setting.

This research was financed by FEDER Funds through “Programa Operacional Factores de Competitividade – COMPETE” and by Portuguese Funds through FCT - “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, within the project PEst-C/MAT/UI0013/2011.