Abstract
In this paper, we present an investigation of quantale algebras (Q-algebras for short) as lattice-valued quantales (Q-quantales for short). First, we prove that the set of all fuzzy ideals of a commutative ring with appropriate operations is a [0, 1]-quantale. Furthermore, we discuss some properties of localic nuclei on Q-algebras, and show that the category of Q-algebras with the quantale structures being frames is a full reflective subcategory of the category of Q-algebras. From this result, we can conclude that the category of L-frames is a full reflective subcategory of the category of L-quantales, where L is a frame. Finally, we build and characterize the Q-quantale completions of a Q-ordered semigroup.
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