An $\mathcal{A}$-semigroup is a numerical semigroup without consecutive small elements. This work generalizes this concept to finite-complement submonoids of an affine cone $\mathcal{C}$. We develop algorithmic procedures to compute all $\mathcal{A}$-semigroups with a given Frobenius element (denoted by $\mathcal{A}(f)$), and with fixed Frobenius element and multiplicity. Moreover, we analyze the $\mathcal{A}(f)$-systems of generators. Furthermore, we study $\mathcal{A}$-numerical semigroups with maximal embedding dimension, fixed Frobenius number and multiplicity, providing an algorithm for their computation and a graphical classification.