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Let    $$\mathbb {N}$$  N be the set of nonnegative integer numbers. A  plane monoid   is a submonoid of    $$(\mathbb {N}^2,+)$$  ( N 2 , + ) . Let  M  be a plane monoid and    $$p,q\in \mathbb {N}$$  p , q ∈ N . We will say that an integer number  n  is  M ( p ,  q )- bounded  if there is    $$(a,b)\in M$$  ( a , b ) ∈ M such that    $$a+p\le n \le b-q$$  a + p ≤ n ≤ b - q . We will denote by    $${\mathrm A}(M(p,q))=\{n\in \mathbb {N}\mid n \text { is } M(p,q)\text {-bounded}\}.$$  A ( M ( p , q ) ) = { n ∈ N ∣ n is M ( p , q ) -bounded } . An    $$\mathcal {A}(p,q)$$  A ( p , q ) - semigroup  is a numerical semigroup  S  such that    $$S= {\mathrm A}(M(p,q))\cup \{0\}$$  S = A ( M ( p , q ) ) ∪ { 0 } for some plane monoid  M . In this work we will study these kinds of numerical semigroups.

Numerical semigroups bounded by the translation of a plane monoid