Kybernetika 47 no. 1, 100-109, 2011

Lattice effect algebras densely embeddable into complete ones

Zdenka Riečanová


An effect algebraic partial binary operation $\oplus$ defined on the underlying set $E$ uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion $\widehat{E}$ of $E$ there exists an effect algebraic partial binary operation $\widehat{\oplus}$ then $\widehat{\oplus}$ need not be an extension of ${\oplus}$. Moreover, for an Archimedean atomic lattice effect algebra $E$ we give a necessary and sufficient condition for that $\widehat{\oplus}$ existing on $\widehat{E}$ is an extension of ${\oplus}$ defined on $E$. Further we show that such $\widehat{\oplus}$ extending ${\oplus}$ exists at most one.


non-classical logics, effect algebras, $MV$-algebras, orthomodular lattices, MacNeille completions


03G12, 06D35, 06F25, 81P10


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