Groups of linear isometries on poset structures

Characterization of Metrics Induced by Hierarchical Posets

“…The important part of this characterization is the canonical part, so we refer to it as the canonical form of a code. In order to describe it, we will first characterize the group of linear isometries of a P -space, which was completely described in [13]. Let us start with some definitions.…”

Posets Admitting the Linearity of Isometries

“…To avoid confusion, let A P denote the order ideal of P generated by A. Assume that a poset P n on [n] and a function f n of F n 2 into F n 2 have been constructed for which (8) supp(f n (u) + f n (v)) Pn = supp(u + v) Pn for all u, v ∈ F n 2 . We define P n+1 as a poset on [n + 1] which contains P n as a subposet, and a function f n+1 :…”

“…Note that Aut P (F n q ) is a subgroup of Iso 0 P (F n q ), and it follows from [3,8] that Aut P (F n q ) = Iso 0 P (F n q ) if P is an anti-chain. Conversely, it is worth studying to determine the posets P which satisfy the property that Aut P (F n q ) = Iso 0 P (F n q ).…”

“…It is called a poset metric. The poset metric spaces have been extensively studied in [1,3,4,5,8]. We briefly introduce basic notions for poset metric on F A subset I of P is called an order ideal (or a down-set) if x ∈ I and y ≤ x imply y ∈ I.…”

“…In [8], Panek et al . determine the structure of such automorphism group Aut P (F n q ), as follows:…”