Proportionally Modular Diophantine Inequalities and Full Semigroups |
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Authors: | JC Rosales JM Urbano-Blanco |
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Institution: | (1) Departamento de Algebra, Universidad de Granada, E-18071 Granada, Spain |
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Abstract: | A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the
type ax mod b ≤ cx. We give a new presentation for these semigroups and we relate them with a type of affine full semigroups.
Next, we describe explicitly the minimal generating system for the affine full semigroups
we are considering. As a consequence, we obtain generating systems for proportionally modular numerical semigroups and we
exhibit several families of these semigroups in terms of their generators. Finally, we use the concept of fundamental gap
to study when a proportionally modular numerical semigroup is symmetric and we propose some open problems. |
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