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Ideals and quotients in lattice ordered effect algebras. (English) Zbl 1004.06009
The authors study the correspondence between ideals and congruences in effect algebras. Kernels of congruences are exactly Riesz ideals. The quotient of a lattice-ordered effect algebra with respect to a Riesz ideal \(I\) is linearly ordered iff \(I\) is a prime ideal, and the quotient is an MV-algebra iff \(I\) is an intersection of prime ideals. Commutators play a key role in orthomodular lattices. As a generalization, the authors define commutators in lattice-ordered effect algebras and study their properties. The quotient with respect to a Riesz ideal \(I\) is an MV-algebra iff \(I\) contains all generalized commutators. Several open questions are formulated at the end of the paper.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06D35 MV-algebras
03G12 Quantum logic
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