In this note, we prove the existence of $E_\leq$-injective hulls in the category $PoSgr_\leq$ of posemigroups and their submultiplicative order-preserving maps; here $E_\leq$ denotes the class of those morphisms $h : A \to B$ for which $h(a_1)...h(a_n)\leq h(a)$ always implies $a_1...a_n\leq a$. The result of this note subsumes the results given by Lambek et al. (2012) and by Zhang and Laan (2014).
Keywords: partially ordered semigroup, quantale, quantic nucleus, injective hull
2010 MSC: 06F05, 06F07, 08B30
Theory and Applications of Categories, Vol. 32, 2017, No. 7, pp 253-257.