### Abstract:

We consider algebraic structures on vector spaces (or chain complexes) V with operations having any number, m, inputs, and any number, n, outputs, including m or n equal to 0. An operation with 0 inputs and n outputs means a choice of an element in the n-fold tensor product of V (for example, the unit of a commutative algebra), an operation with m inputs and 0 outputs means a linear map from the n-fold tensor product of V to the ground field (for example, a linear functional or a pairing), and an operation with 0 inputs and 0 outputs means an element of the ground field, ie a constant (for example, the volume of a manifold as part of an algebra structure its differential forms). The operations may involve a boundary map, so we call the homology classes of the constant operations â€œstructure constantsâ€ . Such an algebraic structure is determined by a certain map. We study this map up to an algebraic version of homotopy, and show, for example, that if the maps defining two algebraic structures are homotopic, then they have equal structure constants. We can also compare algebra structures expressed in different ways on different spaces, and transport (resolved) algebra structures on one space to algebra structures on another space, such that the structure constants only change by an overall scale factor. Given extra structure, we can give explicit formulas for the transported structures. Such extra structure always exists, which allows us to transport a structure on a chain complex to its homology by giving an explicit formula.