Weighted Expansions for Canonical Desingularization Notation Semigroups Strings Semigroup strings with restrictions Ordered semigroup strings with restrictions Strings on rings Indeterminate strings Indeterminate strings with restrictions Restricted degree and order for indeterminate strings Indexing strings Nets Semigroup nets with restrictions Ordered semigroup nets with restrictions Nets on rings Indeterminate nets Indeterminate nets with restrictions Restricted degree and order for indeterminate nets Prechips Isobars for prechips and Premonic polynomials Substitutions Substitutions with restrictions Coordinate nets and Monic polynomials Graded ring of a ring at an ideal Graded ring of a ring Graded rings at strings and nets and the notions of separatedness and regularity for strings and nets Inner products and further notions of separatedness and regularity for strings Inner products and further notions of separatedness and regularity for nets Weighted isobars and weighted initial forms Initial forms for regular strings Initial forms for regular strings and nets Protochips and parachips N support of an indexing string for 2N6 Prescales Derived prescales Supports of prescales Protoscales Inner products for protoscales Scales and isobars Properties of derived prescales Isobars for derived scales Isobars and initial forms for scales Initial forms for scales and regular nets Isobars for protochips Initial forms for protochips and monic polynomials.