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Inexact Residuation in Effect Algebras
Ivan Chajda and Helmut Länger

We study the connections of effect and pseudoeffect algebras to substructural logics which are defined by means of residuated lattices and posets. Avoiding conditionally residuated structures where adjointness holds only for those elements for which the operations appearing in the adjointness condition are defined, we choose to globally define an implication, thereby however relaxing the requirement of its value being unique. We call such an implication inexact. In this approach, conjunction in effect algebras is naturally defined as a partial operation derived from the partial addition operation, and implication is given via the lower cone of two elements instead of their infimum. It turns out that the obtained structure satisfies some kind of adjointness.We also extend our methods to cover pseudoeffect algebras introduced by Dvurečenskij and Vetterlein.

Keywords: Inexact residuation, residuated poset, effect algebra, monotone effect algebra, pseudoeffect algebra, good pseudoeffect algebra

AMS Subject Classification: 03G25, 03B47, 06A11, 06F35

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