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
set row/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 adjA -- the classical Gantmacher--Krein
fluctuating inequalities (1941) for i=j=1. Furthermore, our row/column
operations reveal additional undiscovered fluctuating inequalities for TN
matrices.
The introduced set 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
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].Comment: 2 figures, 39 pages, and minor adjustments in the expositio