  It is known that a positive Boolean function f depending on n variables has
at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show
that f has exactly n + 1 extremal points if and only if it is linear read-once.
The class of linear read-once functions is known to be the intersection of the
classes of read-once and threshold functions. Generalizing this result we show
that the class of linear read-once functions is the intersection of read-once
and Chow functions. We also find the set of minimal read-once functions which
are not linear read-once and the set of minimal threshold functions which are
not linear read-once. In other words, we characterize the class of linear
read-once functions by means of minimal forbidden subfunctions within the
universe of read-once and the universe of threshold functions. Within the
universe of threshold functions the importance of linear read-once func- tions
is due to the fact that they attain the minimum value of the specification
number, which is n + 1 for functions depending on n variables. In 1995 Anthony
et al. conjec- tured that for all other threshold functions the specification
number is strictly greater than n + 1. We disprove this conjecture by
exhibiting a threshold non-linear read-once function depending on n variables
whose specification number is n + 1.
