  A real linear combination of products of minors which is nonnegative over all
totally nonnegative (TN) matrices is called a determinantal inequality for
these matrices. It is referred to as multiplicative when it compares two
collections of products of minors and additive otherwise. Set theoretic
operations preserving the class of TN matrices naturally translate into
operations preserving determinantal inequalities in this class. We introduce
index-row (and index-column) operations that act directly on all determinantal
inequalities for TN matrices, and yield further inequalities for these
matrices. These operations assist in revealing novel additive inequalities for
TN matrices embedded in the classical identities due to Laplace $[$Mem$.$
Acad$.$ Sciences Paris $1772]$ and Karlin $(1968).$ In particular, for any
square TN matrix $A,$ these derived inequalities generalize -- to every
$i^{\mbox{th}}$ row of $A$ and $j^{\mbox{th}}$ column of ${\rm adj} A$ -- the
classical Gantmacher--Krein fluctuating inequalities $(1941)$ for $i=j=1.$
Furthermore, our index-row/column operations reveal additional undiscovered
fluctuating inequalities for TN matrices.
  The introduced index-row/column operations naturally birth an algorithm that
can detect certain determinantal expressions that do not form an inequality for
TN matrices. However, the algorithm completely characterizes the multiplicative
inequalities comparing products of pairs of minors. Moreover, the underlying
index-row/column operations add that these inequalities are offshoots of
certain "complementary/higher" ones. These novel results seem very natural, and
in addition thoroughly describe and enrich the classification of these
multiplicative inequalities due to Fallat--Gekhtman--Johnson $[$Adv$.$ Appl$.$
Math$.$ $2003]$ and later Skandera $[$J$.$ Algebraic Comb$.$ $2004].$
