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Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangiansubmanifold in a symplectic manifold $X$. We construct a family of cyclicunital curved $A_\infty$ structures on differential forms on $L$ with values inthe local system of graded non-commutative rings given by the tensor algebra ofthe orientation local system of $L$. The family of $A_\infty$ structures isparameterized by the cohomology of $X$ relative to $L$ and satisfies propertiesanalogous to the axioms of Gromov-Witten theory. On account of thenon-orientability of $L,$ the evaluation maps of moduli spaces of$J$-holomorphic disks with boundary in $L$ may not be relatively orientable. Todeal with this problem, we use recent results on orientor calculus.

The Fukaya $A_\infty$ algebra of a non-orientable Lagrangian