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Lattices of Torsion Theories for Semi-Automata

In: Semigroups and Their Applications

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  • Wilfried Lex

    (Technische Universität Clausthal, Institut für Informatik)

Abstract

The torsion theory for semi-automata in a general sense or acts, as developed in [4], is further investigated. After recalling some of the basic concepts and results of that theory it is proved by means of a lemma on Galois connections in general that the torsion classes, the torsionfree classes, and the torsion theories of semi-automata of an appropriate category form a complete lattice. These lattices are isomorphic to each other or to the dual; they are considered in more detail: it is shown that the abstract classes of irreducible acts form a complete atomistic Boolean sublattice; further a proof is given that the simple abelian groups are characterized as those groups whose lattice of torsion theories for the corresponding group acts is a pentagon.

Suggested Citation

  • Wilfried Lex, 1987. "Lattices of Torsion Theories for Semi-Automata," Springer Books, in: Simon M. Goberstein & Peter M. Higgins (ed.), Semigroups and Their Applications, pages 83-90, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-3839-7_11
    DOI: 10.1007/978-94-009-3839-7_11
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