States on pseudo MV-algebras and the hull-kernel topology. (English) Zbl 1096.06009

Dvurečenskij, Anatolij; Kalmbach, Gudrun

Atti Semin. Mat. Fis. Univ. Modena 50, No. 1, 131-146 (2002).

Pseudo MV-algebras, introduced by G. Georgescu and A. Iorgulescu [Mult.-Valued Log. 6, No. 1–2, 95–135 (2001; Zbl 1014.06008)] and J. Rachůnek [Czech. Math. J. 52, No. 2, 255–273 (2002; Zbl 1012.06012)], generalize Boolean algebras in two directions: they are “multi-valued” and noncommutative. A. Dvurečenskij [J. Aust. Math. Soc., Ser. A 68, No. 2, 261–277 (2000; Zbl 0958.06006)] proved a natural equivalence between pseudo MV-algebras and unital \(l\)-groups and he described basic properties of states on pseudo MV-algebras [Stud. Log. 68, No. 3, 301–327 (2001; Zbl 0999.06011)]. Three sections of the present paper are devoted to states, extremal states, state morphisms, normal maximal ideals on pseudo MV-algebras (unital \(l\)-groups), and two natural topologies. For example, it is proved (Theorem 3.3) that the extremal states carrying the inherited weak topology (defined on the set of all states) and the normal maximal ideals carrying the hull-kernel topology are homeomorphic. Section 4 deals with states on unital \(l\)-groups. In the last section discrete states on pseudo MV-algebras are studied.

Reviewer: Roman Frič (Košice)

MSC:

06D35

MV-algebras

Citations:

Zbl 1014.06008; Zbl 1012.06012; Zbl 0958.06006; Zbl 0999.06011