I am looking at OEIS sequence A000372 titled: "Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of $n$ variables, number of antichains of subsets of an $n$-set, ..."
I do not understand what is a monotone Boolean function of $n$ variables. I understand the second description in the title, number of antichains of subsets of an $n$-set. Can anyone give me a more clear definition of what is being counted in the first description of the title? A small example (say for $n =3$) would probably help me the most). Is there any "easy" bijection between the two descriptions?
In Boolean algebra we define $\,0<1.\,$ A Boolean function is monotone iff $\,f(x)\,$ satisfies $\,f(x)\le f(y)\,$ for every $\,x\le y\,$. This is generalized to multiple variables by defining, e.g., $\,(x_1,y_1)\le(x_2,y_2)\,$ iff $\,x_1\le x_2\,$ and $\,y_1\le y_2.\,$ Thus $\,f(x,y)\,$ is a monotone function iff $\,f(x_1,y_1)\le f(x_2,y_2)\,$ for every $\,(x_1,y_1)\le(x_2,y_2).\,$
The connection is that a Boolean vector $\,X:=(x_1,x_2,\dots,x_n)\,$ is the indicator function of a subset $\,S\,$ of $\,\{1,2,\dots,n\}\,$ where $\,k\in S\,$ iff $\,x_k=1.\,$