We introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of G. We provide sufficient conditions for Cohen-Macaulayness for closed and nonclosed graphs.Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation.
The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show that the number of relations defining S is greater than or equal to the least number of generators of S minus the rank of the associated group of S. If equality holds, we say S is a complete intersection. The main part of this study is devoted to semigroups of natural numbers generated by 3 elements. These semigroups are complete intersections if and only if they are symmetric in the sense of R. Ap~ry [I]. This result applies to algebraic geometry: An af~ine space-curve C with the parametric equations x = t , y = t D, z = t c, a, b, c natural numbers with greatest common divisor I, is a global idealtheoretic complete intersection,if and only if the semigroup S generated by a, b, c is symmetric. III.
Let J ⊂ I be monomial ideals. We show that the Stanley depth of I/J can be computed in a finite number of steps. We also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps. In both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals.
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