On words that nearly $θ$-commute
The Hamming distance between two equal length words $α, β$ is the number of positions where $α$ and $β$ differ. For $x, y \in Σ^*$ and antimorphic involution $θ$, $x$ $θ$-commutes with $y$, if the Hamming distance between $xy$ and $θ(y)x$ is zero. When the Hamming distance between $xy$ and $θ(y)x$ attains its minimum non-zero value of one, then we say $x$ nearly $θ$-commutes with $y$. This manuscript investigates properties of $x$ and $y$ such that the Hamming distance between $xy$ and $θ(y)x$ is one. We provide a complete characterization of such $x$ and $y$. We introduce a binary relation $R_θ$ on $Σ^*$, where $x R_θy$ holds if and only if $x$ nearly $θ$-commutes with $y$. Finally, for a given $y$, we collect all $x$ such that $x$ nearly $θ$-commutes with $y$, and discuss various properties of this set.