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Riesz ideals in generalized effect algebras and in their unitizations. (English) Zbl 1139.81007
The authors study congruences and ideals of generalized effect algebras, generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices, and generalized MV-algebras. Here “generalized” means that the poset does not have an upper bound; this can be always added by the canonical contruction of unitization. Any Riesz ideal induces a congruence, but the reverse implication does not hold. When any of the above structures is factorized over a Riesz ideal, we obtain a structure from the same class. Only for generalized orthomodular posets such a result is not known. The relations between Riesz ideals of generalized effect algebras and their unitizations are studied. Also illustrative examples are of special interest.

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06C15 Complemented lattices, orthocomplemented lattices and posets
08A55 Partial algebras
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