A new measure of growth for groups and algebras

The notion of a bandwidth growth is introduced, which generalizes the growth of groups and the bandwidth dimension, first discussed by J. Hannah and K. C. O'Meara for countable-dimensional algebras. The new measure of growth is based on certain infinite matrix representations and on the notion of growth of nondecreasing functions on the set of natural numbers. Two natural operations are defined on the set $\Omega^{\star}$ of growths. With respect to these operations, $\Omega^{\star}$ forms a lattice with many interesting algebraic properties; for example, $\Omega^{\star}$ is distributive and dense and has uncountable antichains.
This new notion of growth is applied in order to define bandwidth growth for subgroups and subalgebras of infinite matrices and to study its properties.