Rhodes, John; Silva, Pedro V. Further results on monoids acting on trees. (English) Zbl 1283.20064 Int. J. Algebra Comput. 22, No. 4, 1250034, 69 p. (2012). This paper contains the study of monoids acting on trees and it improves results of J. Rhodes, [in Int. J. Algebra Comput. 1, No. 2, 253-279 (1991; Zbl 0797.20053) with Erratum to diagram, p. 274]. Let \(L\) be the Lyndon-Chiswell function for the semigroup \(S\) with generators \(X\) and let \(L'\) be the unique length function on \(S\) given by the action. Since in [loc. cit.], \(L\) and \(L'\) need not be equal, the authors develop and improve those results such that the length function \(L'\) and Lyndon-Chiswell function \(L\) are equal. To state the main result of this paper, first we recall some definitions and notations. Given a semigroup \(M\), we denote by \(M^I\) the monoid obtained by adjoining a new identity \(I\) to \(M\). A partial transformation monoid is an ordered pair of the form \((X,M)\), where \(X\) is a non-empty set and \(M\) is a submonoid of the monoid \(P(X)\) of all partial transformations of \(X\). If \(M\) is a submonoid of the monoid \(M(X)\) of all full transformations of \(X\), we say that \((X,M)\) is a transformation monoid. Assume that \(X=\bigcup_{i=1}^l(X_i\times\cdots\times X_1)\). For \(i=1,\ldots,l\), we define an equivalence relation \(\equiv_i\) on \(X\) by \[ (x_i,\ldots,x_1)\equiv_i(x'_k,\dots,x'_1)\text{ if \((i\leq j,k\) and }x_i=x'_i,\ldots,x_1=x'_1). \] Given \(\varphi\in P(X)\), we denote by \(\text{dom\,}\varphi\) the domain of \(\varphi\). A mapping \(\varphi\in P(X)\) is said to be sequential if: (SQ1) \(\forall i\in\{2,\ldots,l\}((x_i,\ldots,x_1)\in\text{dom\,}\varphi\Rightarrow(x_{i-1},\ldots,x_1)\in\text{dom\,}\varphi)\); (SQ2) \(\forall i\in\{1,\ldots,l\}\forall (x_i,\ldots, x_1)\in\text{dom\,}\varphi(x_i,\ldots,x_1)\;\varphi\in X_i\times\cdots\times X_1\); (SQ3) \(\forall i\in\{1,\ldots,l\}\forall x,x'\in\text{dom\,}\varphi\;(x\equiv_ix'\Rightarrow x\varphi\equiv_ix'\varphi)\). Given partial transformation monoids \((X_l,M_l),\ldots,(X_1,M_1)\), their wreath product is defined by \[ (X_l,M_l)\circ\cdots\circ(X_1,M_1)=(X_l\times\cdots\times X_1,M_l\circ\cdots\circ M_1), \] where \(M_l\circ\cdots\circ M_1\) consists of all \(\varphi\in P(X)\) satisfying (W1) \(\varphi\) is sequential; (W2) \(\varphi\pi_1\in M_1\); (W3) \((.,a_{i-1},\ldots,a_1)\varphi\pi_i\in M_i\) for all \(i\in\{2,\ldots,l\}\) and \((a_{i-1},\ldots,a_1)\in\text{dom\,}\varphi\). The \(\mathcal L\)-preorder on a monoid \(M\) is defined by \(a\leq_{\mathcal L}b\) if \(a\in Mb\). The Rhodes expansion \(Rh(M)\) of \(M\) is the set of all finite chains of the form \[ m_k\leq_{\mathcal L}\cdots\leq_{\mathcal L}m_1\leq_{\mathcal L}m_0=1 \] with \(k\geq 0\) and \(m_i\in M\). Finally note that \(S(X)=\{\varphi\in M(X):\varphi\) is a permutation of \(X\}\) and \(K(X)=\{\varphi\in P(X):|X\varphi|\leq 1\}\). As the main result of the paper, the authors construct an embedding \[ \varphi\colon Rh(M^I)\to\prod_{i=1}^\delta(X_i,M_i)=\cdots\circ(X_2,M_2)\circ(X_1,M_1) \] into an iterated wreath product of partial transformation semigroups where \(M_{2k+1}\) is a submonoid of \(\{Id_{X_{2k+1}}\}\cup K(X_{2k+1})\) and \(M_{2k+2}\) is a submonoid of \(S(X_{2k+2})\cup K(X_{2k+2})\). Also they prove that \(\varphi\) has an interesting property known as Zeiger property. Reviewer: Behnam Khosravi (Zanjan) Cited in 3 Documents MSC: 20M10 General structure theory for semigroups 20E08 Groups acting on trees 20M20 Semigroups of transformations, relations, partitions, etc. 20M30 Representation of semigroups; actions of semigroups on sets 20B07 General theory for infinite permutation groups Keywords:monoid actions; wreath products; length functions; monoids acting on trees; Rhodes expansions Citations:Zbl 0797.20053 PDFBibTeX XMLCite \textit{J. Rhodes} and \textit{P. V. Silva}, Int. J. Algebra Comput. 22, No. 4, 1250034, 69 p. (2012; Zbl 1283.20064) Full Text: DOI arXiv References: [1] R. Alperin and H. Bass, Combinatorial Group Theory and Topology 111, eds. S. M. Gersten and J. R. Stallings (Princeton University Press, 1987) pp. 265–378. 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