Constructing representations of ordered monoids

A sup-lattice is a complete lattice, a morphism of sup-lattices is a mapping preserving arbitrary suprema. A Quantale is a sup-lattice with an associative multiplication which distrubutes over arbitrary suprema. A quantale module is then a sup-lattice on which the quantale “naturally ” acts. All sup-lattice endo-morphisms of sup-lattice S form a quantale, denoted by Q(S). If Q is a quan-tale and M its left module, then the action provides morphism of quantales Q → Q(M), called representation of Q. Conversely, a representation Q → Q(S) endows S with a Q-module structure. Quantales are obviously residuated lattices, i.e. they admit arrow operations ←, → given by a ≤ b ← c ⇔ a · b ≤ c ⇔ b ≤ c → a. Categorically speaking, the arrows are the right adjoints to the left/right action, respectively. Thus, one could also consider the arrows for the module actions. Similarly to ring theory, it is also possible to derive matrix calculus for quan-tales (modules). (Since we have all suprema, includuing infinitesimal ones, th