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On the other hand, if $t(T)$ is even, there exists such $T$ if and only if $S$ is almost symmetric and different from $\\mathbb{N}$; in this case the type of $S$ is the number of even pseudo-Frobenius numbers of $T$. 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We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \\leq t(T) \\leq 2t(S)+1$. On the other hand, if $t(T)$ is even, there exists such $T$ if and only if $S$ is almost symmetric and different from $\\mathbb{N}$; in this case the type of $S$ is the number of even pseudo-Frobenius numbers of $T$. 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