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Erschienen in:
01.12.2011 | Original Paper
Lattice pseudoeffect algebras as double residuated structures
Erschienen in: Soft Computing | Ausgabe 12/2011
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Abstract
Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.