Some analytic features of algebraic data Original Research Article

Our “algebraic data” merely are the elements of a free algebra and universal matrices (certain families of them able to model objects of a combinatorial interest). Their “analytic features” are the features one can define and study by exploiting an analytic monoid associated with the algebra and among them we consider the “inner” ones, that related with certain subalgebras. Vector space examples of such features are the occurrences of zeroes at certain components of a vector and, in case of a matrix, the notions of sparseness and architecture. Though universal (free) algebras do not have vectors (nor components, nor zeroes), we show that this kind of features together with the efficiency of computing them is universal and has many desired properties, we know from vector spaces. Further features concern diagonality, scalars, flocks and the structure of the universal frame of reference.