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.
Highlights
There are quite a lot of papers investigating injective hulls for algebras
We show that every ordered Σ-algebra has an σ -injective hull, and give its concrete form
Injective hulls for posets were studied by Banaschewski and Bruns ([1], 1967) where they got that the injective hull of a poset is its MacNeille completion
Summary
There are quite a lot of papers investigating injective hulls for algebras. Bruns and Lakser, and independently Horn and Kimura constructed injective hulls of semilattices ([2], 1970 and [3], 1971), and their results were soon applied into S-systems over a semilattice by Johnson, Jr., and McMorris ([4], 1972). Injective hulls were constructed for quantum B-modules [12], as well as for S-semigroups and semicategories [13]. It is natural to study injectivity with respect to homomorphisms that are order-embeddings in the case of ordered algebras. The purpose of this paper is to study injectivity on universal ordered algebras which are many-sorted.
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