Linear read-once and related Boolean functions

It is known that a positive Boolean function [Formula presented] depending on [Formula presented] variables has at least [Formula presented] extremal points, i.e. minimal ones and maximal zeros. We show that [Formula presented] has exactly [Formula presented] 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 functions is due to the fact that they attain the minimum value of the specification number, which is [Formula presented] for functions depending on [Formula presented] variables. In 1995 Anthony et al. conjectured that for all other threshold functions the specification number is strictly greater than [Formula presented]. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on [Formula presented] variables whose specification number is [Formula presented].