In mathematics, if is an associative algebra over some ground field k, then a left associative -bialgebroid is another associative k-algebra together with the following additional maps:an algebra map called the source map, an algebra map called the target map, so that the elements of the images of and commute in , therefore inducing an -bimodule structure on via the rule for ; an -bimodule morphism which is required to be a counital coassociative comultiplication on in the monoidal category of -bimodules with monoidal product . The corresponding counit is required to be a left character (equivalently, the map must be a left action extending the multiplication along ). Furthermore, a compatibility between the comultiplication and multiplications on and on is required. For a