On Presentations of Commutative Monoids

In this paper, all the monoids considered are commutative. If S is a monoid generated by {m1, . . . ,mn}, then S is isomorphic to a quotient monoid of N by the kernel congruence σ of the map φ : N → S, φ(k1, . . . , kn) = ∑n i=1 kimi. Under this setting, a finite presentation for S is a finite subset ρ of Nn×Nn such that the congruence generated by ρ is equal to σ. Rédei proves in [5] that every congruence on 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 is said to be reduced if the quotient monoid N/σ is reduced (i.e. the only unit is the zero element). If M is a subgroup of Z, then we define the set