Associative bialgebroid

In mathematics, if formula_1 is an associative algebra over some ground field "k", then a left associative formula_1-bialgebroid is another associative "k"-algebra formula_3 together with the following additional maps:
an algebra map formula_4 called the source map, 
an algebra map formula_5 called the target map, so that the elements of the images of formula_6 and formula_7 commute in formula_3, therefore inducing an formula_1-bimodule structure on formula_3 via the rule formula_11 for formula_12; an formula_1-bimodule morphism formula_14 which is required to be a counital coassociative comultiplication on formula_3 in the monoidal category of formula_1-bimodules with monoidal product formula_17. The corresponding counit formula_18 is required to be a left character (equivalently, the map formula_19 must be a left action extending the multiplication formula_20 along formula_21). Furthermore, a compatibility between the comultiplication formula_22 and multiplications on formula_3 and on formula_24 is required. For a noncommutative formula_1, the tensor square formula_26 is not an algebra, hence asking for a bialgebra-like compatibility that formula_14 is a morphism of "k"-algebras does not make sense. Instead, one requires that formula_26 has a "k"-subspace formula_29 which contains the image of formula_22 and has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra formula_24. Then one requires that the corestriction formula_32 is a homomorphism of unital algebras. If it is a homomorphism for one such formula_29, one can make a canonical choice for formula_29, namely the so called Takeuchi's product formula_35, which always inherits an associative multiplication via the projection from formula_24. Thus, it is sufficient to check if the image of formula_22 is contained in the Takeuchi's product rather than to look for other formula_29. As shown by Brzeziński and Militaru, the notion of a bialgebroid is equivalent to the notion of formula_39-algebra introduced by Takeuchi earlier, in 1977.
Associative bialgebroid is a generalization of a notion of "k"-bialgebra where a commutative ground ring "k" is replaced by a possibly noncommutative "k"-algebra formula_1. Hopf algebroids are associative bialgebroids with an additional antipode map which is an antiautomorphism of formula_3 satisfying additional axioms. 
The term bialgebroid for this notion has been first proposed by J-H. Lu. The modifier associative is often dropped from the name, and retained mainly only when we want to distinguish it from the notion of a Lie bialgebroid, often also referred just as a bialgebroid. Associative bialgebroids come in two chiral versions, left and right. A dual notion is the notion of a bicoalgebroid. 
There is a generalization, an internal bialgebroid which abstracts the structure of an associative bialgebroid to the setup where the category of vector spaces is replaced by an abstract symmetric monoidal category admitting coequalizers commuting with the tensor product. 