Dihedral and reflexive modules with $\infty$-simplicial faces and dihedral and reflexive homology of involutive $A_\infty$-algebras over unital commutative rings

The concepts of a dihedral and a reflexive module with $\infty$-simplicial faces are introduced. For each involutive $A_\infty$-algebra, the dihedral and the reflexive tensor modules with $\infty$-simplicial faces are constructed. On the basis of dihedral and reflexive modules with $\infty$-simplicial faces that defined by an involutive $A_\infty$-algebra the constructions of the dihedral and the reflexive homology of involutive $A_\infty$-algebras over any unital commutative rings are given. The conception of an involutive homotopy unital $A_\infty$-algebra is introduced. A long exact sequence that connecting the dihedral and the reflexive homology of involutive homotopically unital $A_\infty$-algebras over any unital commutative rings is constructed.