I am wondering whether it makes sense to define a generalized norm (generalized absolute value) of an element of a (new) concrete finite-dimensional unital algebra $A$ over complex numbers. The unital algebra $A$ is not a division algebra, but commutative and a star-algebra (i.e., with an involution sending each algebra element to its unique conjugate counterpart in $A$).
Via the notation of self-conjugation (i.e., $x = x^{*} $), I manage to define the so-called monoid of nonnegative algebra elements $S^\mathit{nonneg} \subset A$, which has a unique zero and identity and is a subalgebra of all self-conjugate elements in $A$.
The monoid $S^\mathit{nonneg} \subset A$ generalizes the monoid of nonnegative real numbers; however, it is only an algebraic lattice, not every pair of elements in $S^\mathit{nonneg}$ are comparable, at least according to my current definitions.
However, given any element $Y$ in $S^\mathit{nonneg} $, there is a unique element $X$ in $S^\mathit{nonneg}$ such that $X\cdot X = Y $. In other words, It makes sense to define a notaton like $\sqrt{Y}$ for all $Y \in S^\mathit{nonneg}$.
If I define a norm $X \mapsto \sqrt{X^{*} \cdot X }$, the norm will be an element of the lattice $S^\mathit{nonneg}$.
Does it make sense, and is there any other generalized norm like that defined in a lattice? To my knowledge, I know nothing like that.
If a norm in a lattice makes sense, my humble opinion is the notation of the notion of metric space can be generalized, but not in a topological approach or under some generalized real-valued norms.