Abstract
Quantum B-algebras as implicational subreducts of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantics for non-commutative algebraic logic. Due to the adjunction between multiplication and its residuals, they can be regarded as the implicational counterpart of partially ordered semigroups. By means of the injective hulls (envelopes) of quantum B-algebras, Rump gave the injective hulls of partially ordered semigroups. In this paper, one part of our work is to give new examples of quantum B-algebras where the lower sets do not form a quantale. Another part of our work is to consider the inverse problem of Rump's way, that is, we can also give the injective hulls of quantum B-algebras in terms of the injective hulls of partially ordered semigroups.
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