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We have a similar concept on numerical semigroups, borrowed from the idea of run.

For a numerical semigroup, a desert is an run of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.

In this setting, for a numerical semigroup $S$, then you can count desserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.

In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.

We have a similar concept on numerical semigroups, borrowed from the idea of run.

For a numerical semigroup, a desert is an run of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.

In this setting, for a numerical semigroup $S$, then you can count deserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.

In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.

We have a similar concept on numerical semigroups, borrowed from the idea of run.

For a numerical semigroup, a desert is an interval of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.

In this setting, for a numerical semigroup $S$, then you can count desserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.

In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.

We have a similar concept on numerical semigroups, borrowed from the idea of run.

For a numerical semigroup, a desert is an run of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.

In this setting, for a numerical semigroup $S$, then you can count desserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.

In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.

We have a similar concept on numerical semigroups, borrowed from the idea of run.

For a numerical semigroup, a desert is an interval of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.

In this setting, for a numerical semigroup $S$, then you can count desserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.

In any case, I my feeling is that you should allow infinite intervals in your definition.

We have a similar concept on numerical semigroups, borrowed from the idea of run.

For a numerical semigroup, a desert is an interval of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.

In this setting, for a numerical semigroup $S$, then you can count desserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.

In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.