Multi-posets in algebraic logic, group theory, and non-commutative topology

The recent discovery that very different types of algebras have a quantale as an injective envelope is analyzed. Multi-posets are introduced as a generic structure that admits an essential embedding into a quantale, explicitly realized as a completion. Effect algebras and their non-commutative extensions, quantum B-algebras, KL-algebras, groups, and various other types of algebras are genuine multi-posets, in the sense that they determine full subcategories. Some structures like partially ordered groups or effect algebras do not carry over to the injective envelope. We provide a criterion for extendability in terms of a generalized archimedean property. Applied to non-commutative topology, multi-posets lead to a symmetric version of quantum spaces as a class of spatial quantales.