ABSTRACT
Chapter 5 Partition Bases and B(1)- Groups Immacolata Caruso Dipartimento di Matematica e Applicazioni, Universita’ Federico II di Napoli, 80100 Napoli, Italy immacaruso@virgilio.it
Clorinda De Vivo Dipartimento di Matematica e Applicazioni, Universita’ Federico II di Napoli, 80100 Napoli, Italy clorinda.devivo@dma.unina.it
Claudia Metelli Dipartimento di Matematica e Applicazioni, Universita’ Federico II di Napoli, 80100 Napoli, Italy claudia.metelli@dma.unina.it
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.3 Partition Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.4 Direct Summands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.5 The Domain of (C,D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.6 Indecomposable Summands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Abstract B(1)-groups are a subclass of the class of Butler groups, the torsionfree quotients of completely decomposable groups. We study the partition structure associated to a B(1)-group G, a lattice-theoretical feature that is behind the direct sum decompositions of G. We determine some of its properties, and give a contribution to the solution of an open problem, that of deciding when the direct sum of two B(1)-groups is a B(1)-group. Keywords: Butler group, B(1)-group, tent, partition lattice, finite algorithm. A.M.S. Classification: 20K15, 06F99, 06B99.
