As we all know, the complete lattice I_D(S) of all D-ideals of a meet-semilattice S is precisely the injective hull of S in the category of meet-semilattices. In this paper, we consider sm-ideals of posemigroups which can be regarded as a generalization of D-ideals of meet-semilattices. Unfortunately, the quantale R(S) of all sm-ideals of a posemigroup S is in general not an injective hull of S. However, R(S) can be seen as a new type of quantale completions of S. Further, we can see that R(S) is also a free object over S in the category PoSgr_v of posemigroups with sm-distributive join homomorphisms.
Keywords: partially ordered semigroup, injective hull, quantale, quantale completion, free object
2010 MSC: 06F05, 06F07, 08B30
Theory and Applications of Categories, Vol. 32, 2017, No. 32, pp 1098-1106.