Bialgebra
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.)
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Bialgebra
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.)
has abstract
En matemáticas, una biálgebra ...... automáticamente una biálgebra
@es
En mathématiques, une bialgèbr ...... t en particulier des bigèbres.
@fr
In mathematics, a bialgebra ov ...... is automatically a bialgebra.
@en
在數學中，域 上的雙代數是兼具 上之結合代數（具單位元）與餘代數的結構，而且這兩種結構彼此相容。最重要的特例之一是霍普夫代數。
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数学において，体 K 上の双代数（そうだいすう，英: bia ...... が有限次元ならいつでも可能である），自動的に双代数になる．
@ja
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1,009,918,123
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En matemáticas, una biálgebra ...... l mismo diagrama conmutativo).
@es
En mathématiques, une bialgèbr ...... t en particulier des bigèbres.
@fr
In mathematics, a bialgebra ov ...... he same commutative diagrams.)
@en
在數學中，域 上的雙代數是兼具 上之結合代數（具單位元）與餘代數的結構，而且這兩種結構彼此相容。最重要的特例之一是霍普夫代數。
@zh
数学において，体 K 上の双代数（そうだいすう，英: bia ...... が有限次元ならいつでも可能である），自動的に双代数になる．
@ja
label
Bialgebra
@de
Bialgebra
@en
Bialgèbre
@fr
Biálgebra
@es
双代数
@ja
雙代數
@zh