On words that nearly $\hat{I}¸$-commute

Abstract

The Hamming distance between two equal length words $\hat{I}\pm ,{\hat{I}}^2$ is the number of positions where $\hat{I}\pm$ and ${\hat{I}}^2$ differ. For $x,y\in \hat{I}{\pounds }^{\ast }$ and antimorphic involution $\hat{I}¸$, $x$ $\hat{I}¸$-commutes with $y$, if the Hamming distance between $xy$ and $\hat{I}¸(y)x$ is zero. When the Hamming distance between $xy$ and $\hat{I}¸(y)x$ attains its minimum non-zero value of one, then we say $x$ nearly $\hat{I}¸$-commutes with $y$. This manuscript investigates properties of $x$ and $y$ such that the Hamming distance between $xy$ and $\hat{I}¸(y)x$ is one. We provide a complete characterization of such $x$ and $y$. We introduce a binary relation $R_{\hat{I}}¸$ on $\hat{I}{\pounds }^{\ast }$, where $x{R}_{\hat{I}}¸y$ holds if and only if $x$ nearly $\hat{I}¸$-commutes with $y$. Finally, for a given $y$, we collect all $x$ such that $x$ nearly $\hat{I}¸$-commutes with $y$, and discuss various properties of this set.