We study the sets of nonnegative solutions of Diophantine inequalities of the form ax mod bpcx with a; b and c positive integers. These sets are numerical semigroups, which we study and characterize. r

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In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid. IntroductionIn this paper, all the monoids considered are commutative. If S is a monoid generated by {m 1 , . . . , m n }, then S is isomorphic to a quotient monoid of N n by the kernel congruence σ of the map ϕ :Under this setting, a finite presentation for S is a finite subset ρ of N n × N n such that the congruence generated by ρ is equal to σ. Rédei proves in [5] that every congruence on N n is finitely generated, and so that every finitely generated monoid is finitely presented.Of all the subsets which generate σ, we are interested in those which have a minimal cardinality and therefore, we will call them presentations of minimal cardinality.A congruence σ on N n is said to be reduced if the quotient monoid N n /σ is reduced (i.e. the only unit is the zero element).If M is a subgroup of Z n , then we define the setwhich is in fact a congruence on N n . Conversely, for a given congruence σ ⊆ N n ×N n , we define the following subgroup of Z n :It is well-known that if σ is a congruence on N n and N n / ∼ Mσ is reduced, then N n /σ is reduced. According to this, we say that a congruence σ is strongly reduced if N n / ∼ Mσ is reduced.

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We introduce the class of the opened modular numerical semigroups (for brevity, OM-semigroups) and obtain formulas for the Frobenius number, the singularity degree and the type of these semigroups. We prove that a numerical semigroup S has a Toms decomposition if and only if S is an intersection of OMsemigroups. We also characterize irreducible proportionally modular numerical semigroups in terms of OM-semigroups.