Enumeration of Numerical Semigroups {0}∪[a,b]∪[c,∞) with Same Embedding Dimension

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Chaiwat Namnak
Chonlada Phanmun
Ekkachai Laysirikul


We consider the set gif.latex?S defined as gif.latex?\{0\}\cup[a,b]\cup[c,\infty) where gif.latex?[a,b] be the set of all integer gif.latex?x such that gif.latex?a\leq&space;x&space;\leq&space;b and gif.latex?[c,\infty) be the set integer gif.latex?y such that gif.latex?c\leq&space;y when gif.latex?a,b and gif.latex?c are positive integers satisfying gif.latex?2&space;\leq&space;a\leq&space;b&space;<&space;c-1. It is known that gif.latex?S is a numerical semigroup if and only if gif.latex?c\leq&space;2a. This research aims to characterize the minimal system of generators for numerical semigroups gif.latex?S and determine the count of numerical semigroups gif.latex?\{0\}\cup[a,b]\cup[c,\infty) that share the same embedding dimension.

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