An effect algebraic partial binary operation $øplus$ 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...

Dually residuated lattice ordered monoids ($DR\ell$-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings ($MV$-algebras, $BL$-algebras) and their non-commutative variants ($GMV$-algebras, pseudo $BL$-algebras). In the paper, lex-extensions and lex-ideals of $DR\ell$-monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.

Bounded commutative residuated lattice ordered monoids ($R\ell$-monoids) are a common generalization of, e.g., $BL$-algebras and Heyting algebras. In the paper, the properties of local and perfect bounded commutative $R\ell$-monoids are investigated.