On rank properties of endomorphisms of finite circular orders

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A mapping alpha from X = {1, 2,..., n} to X is orientation-preserving if the sequence {1 alpha,2 alpha,..., n alpha} is a cyclic permutation of a nondecreasing sequence (with respect to some total order on X). Orientation-preserving mappings can be thought of as preserving a circular order on X. Two partitions of X have the same type if they have identical sizes and numbers of classes. Let tau be a partition with r classes, and let S be the semigroup generated by the set of orientation-preserving mappings from X to X with kernels of same type as tau. We show that the minimum cardinality of a generating set for S is ((n)(r)). Moreover, we characterize all such S generated by their idempotent elements (i.e., s is an element of S such that s(2) = s), and show that the minimum number of idempotent elements required to generate S is ((n)(r)).

Original languageEnglish
Pages (from-to)1237--1250
Number of pages14
JournalCommunications in Algebra
Issue number4
Publication statusPublished - Mar 2006


  • circular order
  • minimum cardinality generating sets
  • minimum cardinality idempotent generating sets
  • orientation-preserving mapping


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