Injective hulls of many-sorted ordered algebras

Abstract

Abstract This paper is devoted to the study of injectivity for ordered universal algebras. We first characterize injectives in the category $\begin{array}{} \displaystyle {\mathsf{OAL}_{{\it\Sigma}}^{\leqslant}} \end{array}$ of ordered Σ-algebras with lax morphisms as sup-Σ-algebras. Second, we show that every ordered Σ-algebra has an σ⩽-injective hull, and give its concrete form.