On the power-set \(Q\)-algebras (Q284662)

A quantale by \textit{C. J. Mulvey} [Suppl. Rend. Circ. Mat. Palermo (2) 12, 99--104 (1986; Zbl 0633.46065)] is a triple \((Q, \bigvee, \&)\) such that \((Q, \bigvee)\) is a complete lattice, \((Q, \&)\) is a semigroup, and for all \(a\in Q,\) \(\{b_i\}_{i\in I \subseteq Q},\) \[ a\& (\bigvee\limits_{i\in I}^{}b_i) = \bigvee\limits_{i\in I}^{}(a\& b_i), (\bigvee\limits_{i\in I}^{}b_i)\& a = \bigvee\limits_{i\in I}^{}(b_i\& a) \] . In an \(R\)-module, if we replace the additional group by a complete lattice and replace the ring with identity by a unital quantale, we can define the new concept \(Q\)-module. Furthermore, we can define \(Q\)-algebras. \(Q\)-algebras form an important class of ordered algebraic structures, which can be regarded as a generalization of quantales and \(Q\)-modules, and play an important role in the study of lattice-valued quantales, lattice-valued frames and stratified lattice-valued topological spaces. In this paper, the authors prove that every \(Q\)-algebra is isomorphic to a quotient \(Q\)-algebra of some power-set \(Q\)-algebra, and investigate some properties of power-set \(Q\)-algebras, and, by means of the relations between ordered semigroups, give a general characterization for the strong homomorphisms between power-set \(Q\)-algebras.