  In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the
log-rank conjecture for communication complexity, up to log n factor, for any
Boolean function composed with AND function as the inner gadget. One of the
main tools in this result was the relationship between monotone analogues of
well-studied Boolean complexity measures like block sensitivity and certificate
complexity. The relationship between the standard measures has been a long line
of research, with a landmark result by Huang (Annals of Mathematics 2019),
finally showing that sensitivity is polynomially related to all other standard
measures. In this article, we study the monotone analogues of standard measures
like block sensitivity (mbs(f)), certificate complexity (MCC(f)) and fractional
block sensitivity (fmbs(f)); and study the relationship between these measures
given their connection with AND-decision tree and sparsity of a Boolean
function. We show the following results: 1) Given a Boolean function $f : \{0,
1\}^{n} \rightarrow \{0, 1\}$, the ratio $fmbs(f^l )/mbs(f^l )$ is bounded by a
function of n (and not l). A similar result was known for the corresponding
standard measures (Tal, ITCS 2013). This result allows us to extend any upper
bound by a well behaved measure on monotone block sensitivity to monotone
fractional block sensitivity. 2) The question of the best possible upper bound
on monotone block sensitivity by the logarithm of sparsity is equivalent to the
natural question of best upper bound by degree on sensitivity. One side of this
relationship was used in the proof by Knop, Lovett, McGuire and Yuan (STOC
2021). 3) For two natural classes of functions, symmetric and monotone, hitting
set complexity (MCC) is equal to monotone sensitivity.
