Iteration of expansions. Unambiguous semigroups
In this paper the author introduces a semigroup expansion which simultaneously enjoys properties of the left and right Rhodes expansions which is used in another paper [reviewed below]. He begins with \(\hat S_ A^{{\mathcal L}}\), the left Rhodes expansion cut down to generators which is defined as \(<(a)| a\in A>_{\hat S^{{\mathcal L}}}\) where \(S=\) and \(\hat S^{{\mathcal L}}\) is the ordinary left Rhodes expansion. \(\hat S^{{\mathcal L}}_ A\) has unambiguous \({\mathcal L}\)-order (i.e. if y and z are \({\mathcal L}\)-above x, then y and z are \({\mathcal L}\)-comparable) and the \(h_{{\mathcal L}}\)-property (i.e. for each \({\mathcal L}\)-class \(L_ 1\) of \(\hat S^{{\mathcal L}}_ A\) there is a finite bound on the length of ascending \({\mathcal L}\)-chains from \(L_ 1)\). \(\hat S^{{\mathcal R}}_ A\) is defined dually. The main idea is to iterate these expansions alternately (iteration of the same expansion has no effect). This leads to two infinite chains of expansions and canonical morphisms the projective limits of which are isomorphic. This expansion is denoted by \(\hat S^+_ A\) and significantly the defining projective limit is finitary, meaning that given \((s_ n)_{n\in\omega }\in\hat S^+_ A\) it is possible to determine the equality of \((s_ n)_{n\in\omega }\) to any other member of \(\hat S^+_ A\) by comparison of the first N members of each sequence (N depending only on \((s_ n)_{n\in\omega })\). This leads to the main theorem which in part says: for any semigroup S, generated by a subset A, there exists a semigroup expansion \(\hat S^+_ A\), generated by a subset of the same cardinality \(| A|\), also denoted by A, and a surmorphism \(\eta:\widehat S^+_ A\twoheadrightarrow S\), injective on A such that (1) \(\hat S^+_ A\) is invariant under the expansions (\({\hat\cdot })^{{\mathcal L}}_ A\) and (\({\hat\cdot })^{{\mathcal R}}_ A\). Hence \(\hat S^+_ A\) has unambiguous \({\mathcal L}\)- and \({\mathcal R}\)- order and has the \(h_{{\mathcal L}}\)- and \(h_{{\mathcal R}}\)-properties. (2) If S is finite [\({\mathcal J}\)-finite above] then \(\hat S^+_ A\) is finite [\({\mathcal J}\)-finite above]. (S is \({\mathcal J}\)-finite above if for each \(s\in S\), \(\{\) \(x\in S| x\geq_{{\mathcal J}}S\}\) is finite.) (3) Each group of S is the pre-image under \(\eta\) of some group of \(\hat S^+_ A\). The inverse image (under \(\eta)\) of a regular \({\mathcal D}\)- class of S, is a regular \({\mathcal D}\)-class and the inverse image of the regular elements of a \({\mathcal J}\)-class is all contained in one \({\mathcal J}\)-class.
length of ascending \({\mathcal L}\)-chains
scientific article; zbMATH DE number 3871665