A categorical characterization of the least Q-quantale completion of Q -ordered semigroups

摘要

The main purpose of this paper is to describe the concrete form of the largest topological fuzzy closure operator on the power-set quantale of a Q-ordered semigroup. Meanwhile, we give a join- and meet-dense basis of the least Q-quantale completion of a Q-ordered semigroup. Moreover, we prove that the least Q-quantale completion is precisely an epsilon(<=)-injective hull of the Q-ordered semigroup in the category Q-Osg(<=) of Q-ordered semigroups and submultiplicative Q-order-preserving mappings. Here epsilon(<=) denotes the class of those morphisms h: S -> T in Q-Osg(<=) for which e(T)(h(a(1)) ... h (a(n)), h (a)) <= e(S)(a(1) ... a(n), a), where es and e(T) are the Q-orders on the Q-ordered semigroups S and T, respectively. (C) 2019 Elsevier B.V. All rights reserved.