ZL-Completions for ZL-Semigroups

In this paper, we generalize a common completion pattern of ordered semigroups to the fuzzy setting. Based on a standard L-completion ZL, we introduce the notion of a ZL-semigroup as a generalization of an L-ordered semigroup, where L is a complete residuated lattice. For this asymmetric mathematical structure, we define a ZL-completion of it to be a complete residuated L-ordered semigroup together with a join-dense L-ordered semigroup embedding satisfying the universal property. We prove that: (1) For every compositive ZL, the category CSL of complete residuated L-ordered semigroups is a reflective subcategory of the category SZL of ZL-semigroups; (2) for an arbitrary ZL, there is an adjunction between SZL and the category SZL→E of weakly ZL-continuous L-ordered semigroup embeddings of ZL-semigroups. By appropriate specialization of ZL, the results can be applied to the DML-completion, certain completions associated with fuzzy subset systems, etc.