Algebraic properties of the category of double quantale modules and its direct limit

This paper is to apply the methods of category theory to investigate double quantale modules properties, the definition of a double quantale module and some relative concepts were introduced. Based on which, some properties of the double quantale module, and the structure of the free double quantale modules generated by a set were obtained. It was proved that the category of double quantale modules is algebraic. Also, the structure of direct limit of the category of double quantale modules was given.