  In this article, we develop an algebraic framework of axioms which abstracts
various high-level properties of multi-qudit representations of generalized
Clifford algebras. We further construct an explicit model and prove that it
satisfies these axioms. Strengths of our algebraic framework include the
minimality of its assumptions, and the readiness by which one may give an
explicit construction satisfying these assumptions. In terms of applications,
this algebraic framework provides a solid foundation which opens the way for
developing a graphical calculus for multi-qudit representations of generalized
Clifford algebras using purely algebraic methods, which is addressed in a
follow-up paper.
