On the monoid of partial isometries of a cycle graph

Abstract

In this paper we consider the monoid DPC n of all partial isometries of a n-cycle graph C n . We show that DPC n is the submonoid of the monoid PORI n of all oriented partial permutations on a n-chain whose elements are precisely all restrictions of the dihedral group of order 2n. For this purpose, a useful characterization of an oriented transformation is established first. Our main aims are to determine the rank and to exhibit a presentation of DPC n . We also describe Green's relations of DPC n and calculate its cardinal. 2020 Mathematics subject classification: 20M20,

Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set

“…They showed that DP(C n ) is an inverse submonoid of the monoid of all oriented partial permutations on a chain with n elements and, moreover, that it coincides with the inverse submonoid of I n formed by all restrictions of a dihedral subgroup of order 2n of S n . This fact motivated the authors of [17] to call DP(C n ) by dihedral inverse monoid on Ω n . Also in [17], it was determined the cardinal and rank of DP(C n ) as well as descriptions of its Green's relations and, furthermore, presentations for DP(C n ).…”

“…This fact motivated the authors of [17] to call DP(C n ) by dihedral inverse monoid on Ω n . Also in [17], it was determined the cardinal and rank of DP(C n ) as well as descriptions of its Green's relations and, furthermore, presentations for DP(C n ).…”

“…, n − 1} ∪ {{1, n}}) with n vertices. The monoid DP(C n ) of all partial isometries of the cycle graph C n was studied by Fernandes and Paulista in [17]. They showed that DP(C n ) is an inverse submonoid of the monoid of all oriented partial permutations on a chain with n elements and, moreover, that it coincides with the inverse submonoid of I n formed by all restrictions of a dihedral subgroup of order 2n of S n .…”