Abstract
The lengths of matrix incidence algebras are studied when their radicals have square zero. All realizable values of the length function are provided for such algebras. In order to obtain this result, a discrete optimization problem is posed and solved. Also, the exact formula of the length is deduced under the additional assumption that the algebra is maximal by inclusion. Moreover, the solution to the length realizability problem is established for matrix incidence algebras with arbitrary radicals under a restriction on the cardinality of the ground field.
Acknowledgments
The author is grateful to his supervisor Professor O.V. Markova for the helpful discussions during the preparation of the paper. The author would like to thank the anonymous referee for the useful comments.
Disclosure statement
No potential conflict of interest was reported by the author.