Some aspects of lattice and generalized prelattice effect algebras

Abstract

Common generalizations of orthomodular lattices and MV-algebras are lattice effect algebras which may include noncompatible pairs of elements as well as unsharp elements. Thus elements of these structures may be carriers of states, or probability measures, when they represent properties, questions or events with fuzziness, uncertainty or unsharpness. Unbounded versions of these structures (more precisely without top elements) are generalized effect algebras which can be extended onto effect algebras. We touch only a few aspects of these structures. Namely, necessary and sufficient conditions for generalized effect algebras to obtain their effect algebraic extensions lattice ordered or MV-effect algebras. We also give one possible construction of pastings of MV - effect algebras together along an MV - effect algebra to obtain lattice effect algebras. In conclusions we give some applications of presented results about sets of sharp elements, direct and subdirect decompositions of lattice effect algebras and about smearings (resp. the existence) of states an probabilities on them. © Springer-Verlag Berlin Heidelberg 2006.