The maximum proportionally modular numerical semigroup with given multiplicity and ratio. (Q633179)

For given positive integers \(a,b,c\) let \(S=S(a,b,c)\) be the additive semigroup of all positive integers \(x\) satisfying \(ax\bmod b\leq cx\), let \(n_1<n_2<\cdots<n_r\) be the unique minimal set of generators of \(S\) and put \(m(s)=a_1\) and \(r(S)=n_2\). The author considers for given co-prime integers \(m,r\) the family \(C(m,r)\) of all semigroups \(S\) with \(m(S)=m\) and \(r(S)=r\). He determines explicitly the maximal (with respect to set inclusion) element of \(C(m,r)\) and finds its minimal generator system, its Frobenius number and the cardinality of its complement.