Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the following intriguing statement (Page 7, section 1.3, end of the first paragraph):
Maybe instead of categories one should study structures with the "Heredity Principle".
In the same paragraph, Gelfand gives an example of a "Heredity Principle" satisfied by the Quasideterminants. Here is the full paragraph (emphasis and sub-paragraphs are mine):
An important problem both in pure and applied mathematics is how to deal with block-matrices. Attempts to find an adequate language for this problem go back to Frobenius and Schur. My colleagues and I think that we found an adequate language: quasideterminants.
Quasideterminants do not possess the multiplicative property of determinants but unlike commutiative determinants they satisfy the more important "Heredity Principle": let $A$ be a square matrix over a division algebra and $(A_{ij})$ a block decomposition of $A$. Consider $A_{ij}$'s as elements of a matrix $X$. Then the quasideterminant of $X$ will be a matrix $B$, and (under natural assumptions) the quasideterminant of $B$ is equal to a suitable quasideterminant of $A$.
Maybe, instead of categories one should study structures with the "Heredity Principle".
Quasideterminants were introduced in
I. Gelfand, S. Gelfand, V. Retakh, R. Wilson, Quasideterminants, https://arxiv.org/abs/math/0208146,
where in Section 3, page 24, a notion of a predeterminant $D_{I,J}(A)$ is introduced, where $A$ is a $n\times n$ matrix and $I,J$ are orderings of $\{1, \ldots, n\}$, followed by the statement:
From the “categorical point of view” the expressions $D_{I, \tilde I}(A)$, where $I=(i_1,i_2,\ldots, i_n)$, $\tilde I=(i_2,i_3,\ldots,i_n,i_1)$, are particularly important.
Earlier in that paper, in a remark on page 11, the authors suggest to generalize quasideterminants to matrices having as entries morphisms in an additive category.
However, no general "Heredity Principle" structures seem to be suggested in any of these sources.
Hence my questions:
- What Gelfand could have meant by his suggestion?
- Any work had been done by anyone along these lines?
