defining and for . Hence -algebras generalize associative algebras. Differential graded algebras Every differential graded algebra has a canonical structure as an -algebra where and is the multiplication map. All other higher maps are equal to . Using the structure theorem for minimal models, there is a canonical -structure on the graded cohomology algebra which preserves the quasi-isomorphism structure of the original differential graded algebra. One common example of such dga's comes from the Koszul algebra arising from a regular sequence. This is an important result because it helps pave the way for the equivalence of homotopy categoriesof differential graded algebras and -algebras. Cochain algebras of H-spaces One of the motivating examples of -algebras comes from the study of H-spaces. Whenever a topological space is an H-space, its associated singular chain complex has a canonical -algebra structure from its structure as an H-space. Example with infinitely many non-trivial mi Consider the graded algebra over a field of characteristic where is spanned by the degree vectors and is spanned by the degree vector . Even in this simple example there is a non-trivial -structure which gives differentials in all possible degrees. This is partially due to the fact there is a degree vector, giving a degree vector space of rank in . Define the differential by and for where on any map not listed above and . In degree , so for the multiplication map, we have And in the above relations give When relating these equations to the failure for associativity, there exist non-zero terms. For example, the coherence conditions for will give a non-trivial example where associativity doesn't hold on the nose. Note that in the cohomology algebra we have only the degree terms since is killed by the differential . Properties Transfer of A∞ structure One of the key properties of -algebras is their structure can be transferred to other algebraic objects given the correct hypotheses. An early rendition of this property was the following: Given an -algebra and a homotopy equivalence of complexes , then there is an -algebra structure on inherited from and can be extended to a morphism of -algebras. There are multiple theorems of this flavor with different hypotheses on and , some of which have stronger results, such as uniqueness up to homotopy for the structure on and strictness on the map . Structure Minimal models and Kadeishvili's theorem One of the important