Let N d be the d-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid S ⊆ N d such that H(S) = N d \ S is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of H(S). In particular, we describe relaxed monomial orders on N d , define the Frobenius element of S with respect to a given relaxed monomial order, and show that the Frobenius element of S is independent of the order if the generalized numerical semigroup is irreducible.

We give a characterization on the sets A ⊆ ℕd such that the monoid generated by A is a generalized numerical semigroup (GNS) in ℕd. Furthermore we give a procedure to compute the hole set ℕd \ S, where S is a GNS, if a finite set of generators of S is known.

In this paper, we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture, we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of N d with finite complement in N d . These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in N d we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in N d we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). It was used to compute the number of such semigroups, and its implementation allowed us to compute (for various dimensions) the number of numerical semigroups for genus that had not been attained before.2010 Mathematics Subject Classification. 20M14, 05A15, 11D07,20-04.

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