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Starting from a biased definition of a properad, we describe explicitlyalgebras over the cobar construction of a properad. Equivalent description interms of solutions of generalized master equations, which can be interpreted ashomological differential operators, are explained from the properadic point ofview. This is parallel to the Barannikov's theory for modular operads. Inaddition to well known IBL-homotopy algebras, the examples include theirassociative analogues, which we call $IBA$-homotopy algebras, and a combinationof the above two.

Properads and Homotopy Algebras Related to Surfaces