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This work focuses on the combinatorial properties of glued semigroups and provides its combinatorial characterization. Some classical results for affine glued semigroups are generalized and some methods to obtain glued semigroups are developed.

Let

Let

The main goal of this work is to study the semigroups which result from the gluing of other two. This concept was introduced by Rosales in [

As seen, glued semigroups can be determinated by the minimal generating sets of

Furthermore, some methods which require linear algebra and integer programming are given to obtain examples of glued semigroups.

The content of this work is organized as follows. Section

A binomial of

With the above notation, the semigroup

Let

For each

Let

It is clear that if

The following proposition generalizes [

The semigroup

Assume that

Conversely, suppose that there exists

If

If

Using that

the binomial

The case

We conclude that

From the above proof it is deduced that given the partition of the system of generators of

Glued semigroups by means of nonconnected simplicial complexes are characterized. For any

The following result shows an important property of the simplicial complexes associated with glued semigroups.

Let

Suppose that there exists

Since

If

If

then

The case

We now describe the simplicial complexes that correspond to the

Let

If there exist

Conversely, let

The following lemma is a combinatorial version of [

Let

The order

Assume that there exists a mixed monomial

After examining the structure of the simplicial complexes associated with glued semigroups, we enunciate a combinatorial characterization by means of the nonconnected simplicial complexes

The semigroup

For all

There exists a unique

For all

Besides, the above

If

Conversely, by hypotheses 1 and 3, given that

From Theorem

Let

The ideals

The element

For all

Suppose that

Conversely, suppose that

if

otherwise,

We conclude that

The following example taken from [

Let

Using the appropriated notation for the indeterminates in the polynomial ring

Nonconnected simplicial complexes associated with

In this section, an algorithm to obtain examples of glued semigroups is given. Consider

The following proposition shows that the semigroup

The semigroup

Use that

Because

The semigroup

Suppose that the set of generators

If

If

If

If

We have just proved that

From the above result we obtain a characterization of glued semigroups:

Let

From Example

Take

The semigroup

With the conditions fulfilled by

The following corollary gives the explicit conditions that

The semigroup

there exist

It is trivial by the given construction, Corollary

Therefore, to obtain an affine glued semigroup it is enough to take two affine semigroups and any solution

Let

Second, by Corollary

We now take

All glued semigroups have been computed by using our program

The authors declare that there is no conflict of interests regarding the publication of this paper.

J. I. García-García was partially supported by MTM2010-15595 and Junta de Andalucía group FQM-366. M. A. Moreno-Frías was partially supported by MTM2008-06201-C02-02 and Junta de Andalucía group FQM-298. A. Vigneron-Tenorio was partially supported by Grant MTM2007-64704 (with the help of FEDER Program), MTM2012-36917-C03-01, and Junta de Andalucía group FQM-366.

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