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We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gröbner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibles for the monoid of multiplicities on an arbitrary matroid. [ABSTRACT FROM AUTHOR]

We discuss the notion of Poincaré duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincaré duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang–Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated. [Copyright &y& Elsevier]

The purpose of this article is to give criteria for the Gorenstein property of the blowup of the spectrum of a Gorenstein local ring A along an ideal I⊂⃒A by means of the coefficient ideal of I. Moreover, we investigate the conductor ideal and the Gorenstein property of the fiber cone. [ABSTRACT FROM AUTHOR]

In this article we investigate the Gorenstein property of the associated graded ring GA(I) of an ideal I in a Gorenstein local ring (A,m) of positive dimension. We especially concentrate on the case where I is m-primary. Assuming that the Rees algebra of I is Cohen–Macaulay we then give necessary and sufficient conditions for the Gorensteiness of GA(I) in terms of the Hilbert coefficients of I. [Copyright &y& Elsevier]