BLOCK-FINITE EFFECT ALGEBRAS AND THE EXISTENCE OF STATES
Zdenka Riečanová · Demonstratio Mathematica · 2003
Lattice effect algebras generalize orthomodular lattices and MV-algebras in the quantum or fuzzy probability theory.Every lattice effect algebra E is a union of its maximal MV-effect subalgebras called blocks of E. We show that an Archimedean lattice effect algebra with exactly two blocks is either a horizontal sum of two blocks or (up to isomorphism) a direct product of an MV-effect algebra and a horizontal sum of two blocks.Further, every complete effect algebra with nontrivial center and finitely many blocks is isomorphic to a direct product of an MV-effect algebra M (it may be M = {0}) and finitely many effect algebras with trivial centers and at least two blocks each.As corollaries we obtain the existence of states or order-continuous subadditive states (probabilities) on some complete or Archimedean effect algebras with nontrivial center and finitely many blocks.