Hammocks and the algorithms of Zavadskiĭ (Q1922487)

Given a finite partially ordered set (poset) \(S\), denote by \(\ell(S,k)\) the category of finite-dimensional representations of \(S\) over \(k\), and denote by \({_\infty\ell(S,k)}\) the preprojective component of the Auslander-Reiten quiver of \(S\). A combinatorial notion of a left hammock was introduced by \textit{C. M. Ringel} and \textit{D. Vossieck} [in Proc. Lond. Math. Soc., III. Ser. 54, 216-246 (1987; Zbl 0621.16031)], where, among others, the following two statements were proved. For every thin left hammock \(H\), there is a unique up to isomorphism poset \(S(H)\) satisfying \(H\cong{_\infty\ell(S(H),k)}\) (independent of the field \(k\)), and for every poset \(S\), the preprojective component \({_\infty\ell(S,k)}\) is a thin left hammock. Let \(a,b\) be incomparable elements of a poset \(S\), and let \(P(a)\) and \(Q(b)\) be the associated projective and injective objects, respectively, in \({_\infty\ell(S,k)}\). For \(H={_\infty\ell(S,k)}\), the author constructs a thin left hammock \(\vphantom H^{\vphantom\diamondsuit}_aH_b^\diamondsuit\) and shows that the associated poset \(S(\vphantom H^{\vphantom\diamondsuit}_aH_b^\diamondsuit)\) can be obtained from \(S\) by a finite sequence of applications of \textit{A. G. Zavadskij}'s differentiation algorithm [Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 5, 1007-1048 (1991; Zbl 0809.16011)]. This result is a continuation of the work [in op. cit.] on connections between left hammocks and representations of posets.