Piochi, Brunetto Permutability of normal extensions of a semilattice. (English) Zbl 0737.20031 Semigroup Forum 43, No. 2, 151-162 (1991). A semigroup \(S\) is called \(m\)-permutable (we write \(S\in P_ m\)) if, for the integer \(m\), every product of \(m\) elements can be re-ordered, and \(S\) is called permutable if \(S\) is \(m\)-permutable for some \(m\) (we write \(S\in P\)). An inverse semigroup \(S\) is called a normal extension of an inverse semigroup \(K\) if \(K\) is isomorphic to a normal (meaning full and self-conjugate) subsemigroup of \(S\). It is proved that every normal extension \(S\) of a finite semilattice \(E\) by an inverse permutable semigroup \(Q\) is permutable, and a bound on the permutability index \(k\) is given in terms of that of \(Q\) and certain height related features of \(E\). If \(E\) is a chain and \(Q\in P_ m\) then \(S\in P_ m\) and \(S\) is a semilattice of groups. Some properties of permutable groups are stated and extended to inverse semigroups: for instance, if the semilattice \(E\) of an \(E\)-unitary inverse semigroup \(S\) is finite then \(S\in P\) if and only if the greatest group image \(S/\sigma\in P\). Reviewer: P.M.Higgins (Colchester) Cited in 1 Document MSC: 20M05 Free semigroups, generators and relations, word problems 20M18 Inverse semigroups Keywords:permutation property; inverse semigroup; normal extension; finite semilattice; inverse permutable semigroup; permutability index; semilattice of groups; permutable groups PDFBibTeX XMLCite \textit{B. Piochi}, Semigroup Forum 43, No. 2, 151--162 (1991; Zbl 0737.20031) Full Text: DOI EuDML References: [1] Curzio, M., P. Longobardi and M. Maj,Su di un problema combinatorio in teoria dei gruppi, Atti Acc. Lincei Rend. fis. VIII74 (1983), 136–142. · Zbl 0528.20031 [2] Curzio, M., P. Longobardi, M. Maj, D.J.S. Robinson,A permutational property of groups, Arch. Math.44 (1985), 385–389. · Zbl 0544.20036 [3] Howie, J. M.,An introduction to semigroup theory, Academic Press, London, 1976. · Zbl 0355.20056 [4] Justin, J. and G. Pirillo,Comments on permutation property for semigroups, Semigroup Forum39 (1) (1986), 109–112. · Zbl 0665.20028 [5] Petrich, M.,Inverse Semigroups, Wiley & Sons, New York, 1984. · Zbl 0546.20053 [6] Piochi, B.,Solvability in inverse semigroups, Semigroup Forum34 (3) (1986), 287–303. · Zbl 0604.20058 [7] Piochi, B.,On permutation property in finite semigroups, C. R. Acad. Sci. Paris, Série I309 (1989), 969–974. · Zbl 0691.20045 [8] Restivo, A. and C. Reutenauer,On the Burnside problem for semigroups, J. Algebra89 (1984), 102–104. · Zbl 0545.20051 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.