On Numerical Semigroups with Fixed Quotient

Let $\Delta$ be a numerical semigroup and let $d\ge 2$ be an integer. We study the fiber of the quotient map \(S\mapsto S/d\) over $\Delta$. We describe its elements as semigroups of the form $\langle X\rangle+d\Delta$, for suitable finite sets $X\subseteq\Delta$, and then analyze explicit and computable regions of this fiber. In particular, we introduce a family $\Delta_d(a)$ of multiples with prescribed quotient and compute its generators, classical invariants, Ap\'ery sets, and presentations. We also show that this construction preserves Wilf's inequality and controls the depth. Finally, we introduce the $\mathcal{M}_d(\Delta)$-rank, determine its maximal value in the fiber, relate it to the ordinary embedding dimension, characterize the rank-one elements, and give closed formulas for their Frobenius-type invariants and pseudo-Frobenius numbers.