ON THE LOOMIS–SIKORSKI THEOREM FOR MV-ALGEBRAS WITH INTERNAL STATE

Abstract

Abstract In Flaminio and Montagna [‘An algebraic approach to states on MV-algebras’, in: Fuzzy Logic 2 , Proc. 5th EUSFLAT Conference , Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206; ‘MV-algebras with internal states and probabilistic fuzzy logic’, Internat. J. Approx. Reason. 50 (2009), 138–152], the authors introduced MV-algebras with an internal state, called state MV-algebras. (The letters MV stand for multi-valued.) In Di Nola and Dvurečenskij [‘State-morphism MV-algebras’, Ann. Pure Appl. Logic 161 (2009), 161–173], a stronger version of state MV-algebras, called state-morphism MV-algebras, was defined. In this paper, we present the Loomis–Sikorski theorem for σ -complete MV-algebras with a σ -complete state-morphism-operator, showing that every such MV-algebra is a σ -homomorphic image of a tribe of functions with an internal state induced by a function where all the MV-operations are defined by points.