Boolean functions Boolean function f(x1,x2,x3):
If
f(x1,x2,x3)= TRUE
then
f(TRUE,x2,x3)= TRUE
f(x1,TRUE,x3)= TRUE
f(TRUE,TRUE,x3)= TRUE
f(x1,x2,TRUE)= TRUE
f(TRUE,x2,TRUE)= TRUE
f(x1,TRUE,TRUE)= TRUE
f(TRUE,TRUE,TRUE)= TRUE
For 2 arguments there is 6 such functions. How many such functions for N arguments?
What is the class of these function called?

Edit:
For N=2 variables, these are the functions:
A   B|  F1  F2  F3  F4  F5  F6
0   0|  0   0   0   0   0   1
0   1|  0   0   0   1   1   1
1   0|  0   0   1   0   1   1
1   1|  0   1   1   1   1   1

F1(a,b) = False
F2(a,b) = A & B
F3(a,b) = A
F4(a,b) = B
F5(a,b) = A or B
F6(a,b) = True
 A: Firstly some background issues,
Boolean function is same just like other mathematical function f: (B^n)--->B , provided B={0,1} is a boolean domain and n is non-negative # .
Boolean function of 1 variable can have 2 values 0 and 1.
and with 0 var either it'll be 0 or 1, completely constant with no logical operations.
Similarly a boolean function of n var can have 2^n combinations of values.
eg:- 3 var boolean functions can have 2^3 elements (all from 0 to 7 in binary.... 000,001,010,011,100,101,110,111)
now monotone boolean function . First understand the word monotone, which means either it is going up (increasing) or down (decreasing).
(1) shows increasing ; 2 shows decreasing; 3 is a constant y(that line is parallel to x axis) ; 4th  shows increment and decrement both
Now coming to MBF (Monotone Boolean Function) : Function in which the value of the function follows the value of arguments. Consider a monotonic function fm and <x1 , x2 , ....., xn> as well as <y1,  y2, .... , yn> as the sequence of the truth values :
if  <x1 , x2 , ....., xn> <= <y1,  y2, .... , yn>
then fm(x1 , x2 , ....., xn) <= fm(y1,  y2, .... , yn) .
To make this more sensible , consider F <T .
distributive lattice is a lattice which follows the distributive property

   1. (x+(y.z))=((x.y)+(x.z))
   2. (x.(y+z))=((x.y)+(x.z))


Now look at the boolean lattice below in the image . Boolean lattice is nothing but a distributive lattice with 0 and 1 in which every node / element has a complement (000 and 111 belongs to same lattice ). (0 is not equal to 1).
3-tuple boolean lattice
Now answer to your question
There's no function from F1 to F6 which satisfies this , except for F5 only. but if you consider 4- var function (input of values from the combination of A and B ) then from F1 to F6 they all are a part of MBFunction.

Furthermore , MBFs can't be concluded without looking at the logical
expressions . Therefore for the n-arguments you'll have to state the
logical expression for it. Although i must tell you , the # of MBFs
in an n var is called nth Dedekind # .  Its' calculation is still a
challenge for all of us. For upto 8 variables only this is known.
for upto n=6 , we know the in-equivalent MBFs.

forgive me for the handmade images,in Img 1 , 3rd graph looks more of a non-constant but consider that line as parallel to x-axis
# means number
(edit-reference)
    for reference, visit:
 https://www.sfu.ca/~tstephen/Papers/r7.pdf

Thanks.
