On algebraic and arithmetic properties of monoids of product-$K$ sequences

Let $G$ be a group and $K$ be a normal subgroup of $G$. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed, and the order is disregarded. A product-$K$ sequence is a sequence whose terms can be ordered such that their product in $G$ belongs to $K$. The set $\mathcal B_K (G)$ of all product-$K$ sequences over $G$ forms a monoid, called the monoid of product-$K$ sequences, under the operation of sequence concatenation. In this paper, we investigate the algebraic and arithmetic properties of the monoid $\mathcal B_K (G)$. Among our main results, we provide precise characterizations of when the monoid $\mathcal B_K (G)$ satisfies key properties, namely being a (transfer) Krull, seminormal, or (half-)factorial. Our results generalize existing frameworks, making them applicable to both the classical abelian and the more recently developed non-abelian settings.