On an embedding theorem for a Möbius-function (Q1898562)

Let \((P,R)\), \(R \subseteq P \times P = P^2\),be an ordered set (here and henceforth are meant partially ordered locally finite families of sets). Its Möbius function \(\mu_R\) is defined by Hall's formula: \[ \mu_R (x,y) = \sum_{k \geq 0} (-1^k) \mid L(R, x, y, k)|,\tag{1} \] where \(x, y \in P\) are arbitrary, \(L(R,x,y,k)\) is the set of chains in \((P,R)\) between \(x\) and \(y\) of length \(k\), and \(|X|\) is the number of elements of the set \(X\). The calculation of \(\mu_R\) for a concrete \((P,R)\) sometimes encounters difficulties. In two papers [``On the foundations of a real Möbius- theory'' (Russian), Prepr. Inst. Phys. Sib. Div. USSR Acad. Sci., Krasnoyarsk (1979); Dokl. Akad. Nauk SSSR 260, 40-43 (1981; Zbl 0486.05009)], \textit{B. S. Stechkin} suggested an approach for calculating the Möbius function by step-by-step variation of the carrier of \((P,R)\) -- the so-called embedding theorems. The present paper (together with an earlier one [the author, Diskretn. Mat. 3, No. 2, 121-127 (1991; Zbl 0744.06004)]) is an attempt of further development of this approach.