Several Combinatorial And Algebraic Problems Related To An Order

Order is important in combinatorics and algebra. Many studies related to al-gebraic properties, such as free resolution, shellability and so on, have something todo with a special order. Based on the studies of posets, in this paper we have hada research on several important algebraic aspects as is shown in the following sevenparts.Chapter one is an introduction and contains some preliminaries about the zero-divisor graph of posets and the study on some topic of combinatorial commutativealgebra relative to posets.Chapter two is about the zero divisor graphs of posets, in which we describe therelationship of a poset and its zero divisor graph.Chapter three focuses on a special class of zero-divisor graphs called the Booleangraphs, denoted by BX, which is the zero-divisor graph of the power set of X. We alsostudy a kind of ring, whose annihilating-ideal graph corresponds to the blow-up of aBoolean graph.In chapter four, we discuss a class of ideal called Lyubeznik ideal, whose Lyubeznikresolution is a minimal free resolution. In order to judge whether an ideal is a Lyubeznikideal, the essential point is to fnd a total order satisfying some special conditions. Focuson this point, we fnd some classes of Lyubeznik ideals.In the ffth part, another class of interesting ideal is studied, which is the socalled f ideal. One of the most important concept related to the study of f ideal isthe upper generated set and the lower cover set, which come from the partially orderof divisor relationship on the monomials. After introducing the concept of perfectsets, we characterize f ideals by a simple and direct way, and then give a completecharacterization of the homogeneous f ideals of degree2. We also studied the unmixedproperty of f-ideals.The objects studied by us in the sixth part are several classes of monomial ideals,such as Borel type, Borel fxed, strongly stable, lexsegment ideal and so on. We checkthe invariance of them under some classical ideal operations, such as sum, intersection,product, colon, integral closure, the kthsymbolic power and so on. Note that each ofthese classes of monomial ideals is related to some kind of monomial order.The last chapter is about the spanning complex of a fnite simple graph. We havea research on its shellability and the property of having linear quotients. By fnding a proper total order on the set of facets, we show by a fundamental method that thespanning complex of any graph is shellable and hence is Cohen-Macaulay, and we alsoshow that its facet ideal has linear quotients.