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1. Since the introduction by Weispfenning [We92] of Comprehensive Grobner bases and systems (CGB and CGS), different algorithms have been developed for obtaining a CGS. Let K be a computable field and K an algebraically closed extension of K. Given a generating set {p1, · · · , ps} of the parametric ideal I ⊂ K[a][x], where a = a1, . . . , am are the parameters and x = x1, . . . , xn the variables, and a monomial order x in the variables, a CGS is a set of pairs (Si, Bi) with Si ⊂ K (called segments) and Bi ⊂ K[a][x] (called bases) that specialize to a Grobner basis in K[x] for all points a ∈ Si. Probably the fastest family of algorithms are based [Ka97] on the stability of Grobner bases in K[x, a] with a product order with x a by specialization. Sato and Suzuki [SuSa06], Nabeshima [Na07] and Kapur-Sun-Wang [KaSuWa10] have successively improved the most popular of these algorithms. The objective of them is to obtain a CGS without any more requirements, and they can benefit from the existing fast implementations of ordinary Grobner bases computations. Moreover, the last version in [KaSuWa10] already obtains a disjoint CGS. 2. Another family of algorithms [Mo02, We03, MaMo09], have as objective obtaining a canonical CGS with good properties for applications. These algorithms work in K[a][x] and use only the x as variables, but on the counterpart they need specific algorithms for Grobner bases computations. Based on Wibmer’s Theorem [Wi07] we have introduced the canonical Grobner cover [MoWi10] and implemented it in the Singular grobcov.lib library whose beta version and an executable tutorial can be downloaded at [Mo-web]. The canonical Grobner cover consist of a set of triplets {(lpp1, B1, S1), . . . , (lppr, Br, Sr)} with the following properties

Let there be given a set of monomials in n variables and some order relations between them. The following fundamental problem of monomial ordering is considered. Is it possible to decide whether these ordering relations are consistent and if so to extend them to an admissible ordering for all monomials? The answer is given in terms of the algorithm MACOT which constructs a matrix of so called cotes which establishes the desired ordering relations. The main area of application of this algorithm, i.e. the construction of Gröbner bases for different orderings and of universal Gröbner bases is treated in the last section.

In this paper we present two algorithms for irreducible decomposition of monomial ideals. We first use staircase structures to study the monomial ideals. We generalize the shifting degrees rule from two variables to three variables and then arbitrary case. With the aid of monomial tree representation, a new algorithm for irreducible decomposition of monomial ideals is provided. For the second method, we associate a monomial ideal with a Scarf complex, which is introduced by Herbert Scarf. Every facet of the Scarf complex corresponds to an irreducible component, and vice versa. Milowski(2004) developed a method to enumerate all the facets based on reverse search by exchanging one monomial at a time. We define a facet graph where nodes are facets and there is an edge from a facet F A to another facet F B if F B can be obtained from F A by exchanging one vertex. We give a new generic deformation called ordinal deformation and prove that the facet graph of an ordinally generic monomial ideal is strongly connected. This yields a simpler and more efficient enumeration algorithm.