Multigraded commutative algebra of graph decompositions

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K ₄-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.     

Rings and Gödel algebras

In this paper, we study those rings whose semiring of ideals can be given the structure of a Gödel algebra. Such rings are called Gödel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Gödel ring R is a subdirect product of prime Gödel rings R _i , and the Gödel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into ${{\prod_{i}}Id(R_{i})}$ , where each Id(R _i ) is the algebraic lattice of ideals of R _i that can be equipped with the structure of a Gödel algebra. We see that the mapping associating to each Gödel ring its Gödel algebra of ideals is functorial from the category of Gödel rings with epimorphisms into the full subcategory of frames whose objects are Gödel algebras and whose morphisms are complete epimorphisms.     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

On the n—nilpotency

In this paper we prove that, for every integer n ≥ 1, d ≥ m ≥ 2, the sum of m n- nilpotent ideals of an algebra may not be d- nilpotent. This leads to a similar result in group theory; the product of m n- nilpotent normal subgroups may not be d- nilpotent. We apply these results to solve Rozhkov’s question by constructing some n- finite, infinite p- groups generated by n + 1 conjugates.     

Ideal Structure and Stable Rank of $${\mathbb {C} e+ \ell^2(I)}$$ with the Hadamard Product

Let I be any index set. We consider the Banach algebra \mathbb C e+ l²(I){\mathbb {C} e+ \ell^2(I)} with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For I=\mathbb N{I=\mathbb {N}} we also characterize all prime z-ideals in this Banach algebra.     

Complexity of multiplication in commutative group algebras over fields of characteristic 0

Let rkA denote the bilinear complexity (also known as rank) of a finite-dimension associative algebra A. Algebras of minimal rank are widely studied from the point of view of bilinear complexity. These are the algebras A for which the Alder-Strassen inequality is satisfied as an equality, i.e., rkA = 2dimA ? t, where t is the number of maximum two-sided ideals in A. It is proved in this work that an arbitrary commutative group algebra over a field of characteristic 0 is an algebra of minimal rank. The structure and precise values of the bilinear complexity of commutative group algebras over a field of rational numbers are obtained.     

A characterization of minimal left or right ideals of matrix algebras

The algebra of m×m matrices over real numbers, complex numbers or quaternions is equipped with the standard inner product Re tr(xy ^*) Work done in part during the Summer Research Institute of the Canadian Mathematical Congress, 1971.). It is easy to see that the left ideals of this algebra have the property that the projections of unitary matrices on them have constant length. of course, the right ideals have the same property. This property and a condition on dimension give a characterization of minimal left or right ideals of this algebra. We use this characterization to determine all orthogonal transformations of this algebra which preserve unitary matrices.     

On traceable factor representations of crossed products

In this paper we give a complete classification of the traceable factor representations of the C¹-algebra which is the crossed product of the gauge group of automorphisms with the fermion algebra. Besides the type I representations, this algebra has an uncountable family of type II_∞ traceable factor representations. Unlike the fermion algebra, it has no finite factor representations. We present a similar analysis of the crossed product of SU(2) with the fermion algebra, where the action is the natural product action.     

Irreducible highest weight representations of the simple n-Lie algebra

A. Dzhumadil??daev classified all irreducible finite dimensional representations of the simple n-Lie algebra. Using a slightly different approach, we obtain in this paper a complete classification of all irreducible, highest weight modules, including the infinite-dimensional ones. As a corollary we find all primitive ideals of the universal enveloping algebra of this simple n-Lie algebra.     

Countability properties of the ideal space of a Banach algebra and d-algebras

The space Id(A) of closed two-sided ideals of a Banach algebra A carries the weak topology. IfA is separable Id(A) is expected to satisfy topological countability conditions. Some results concerning first and second countability are achieved.     

Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.     

On the closed Lie ideals of certainC *-algebras

In this article we determine the closed Lie ideals of a uniformly hyperfiniteC ^*-algebra, and of the tensor product of such an algebra withC(X), the space of continuous functions on a compact, Hausdorff space. This is done by localizing the Lie ideals in algebras of the form , where is an algebra over a field of characteristic not equal to 2.This research is partially supported by NSERC (Canada)     

The algebra of one-sided inverses of a polynomial algebra

We study in detail the algebra S_n in the title which is an algebra obtained from a polynomial algebra P_n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P_n. The algebra S_n is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of S_n is 2n; but the weak and the global dimensions of S_n are n. The prime and maximal spectra of S_n are found, and the simple S_n-modules are classified. It is proved that the algebra S_n is central, prime, and catenary. The set I_n of idempotent ideals of S_n is found explicitly. The set I_n is a finite distributive lattice and the number of elements in the set I_n is equal to the Dedekind number d_n.     

布尔代数的直觉T-S模糊子代数及理想

在布尔代数中引入了的直觉T-S模糊子代数和直觉T-S模糊理想的概念,给出了布尔代数的直觉T-S模糊子代数的两个等价定义,进一步讨论了它们的性质.证明了布尔代数的两个直觉T-S模糊子代数(理想)的模交与直积也是直觉T-S模糊子代数(理想).     

Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation type

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6.     

The Shilov boundary and the Gelfand spectrum of algebras of generalized analytic functions

Let S be a discrete abelian cancellation semigroup with identity. We consider an algebra of generalized analytic functions defined on the semigroup $ \hat S $ of semicharacters of S that is wider than the Arens-Singer algebra. We show that the strong boundary and the Shilov boundary of this algebra are unions of maximal subgroups of $ \hat S $ . If S contains no nontrivial simple ideals, then both boundaries coincide with the character group of S. In this case we calculate the Gelfand spectrum of the algebra under consideration.     

On the Maximal and Minimal Left Ideals of Certain Banach Algebras on Locally Compact Groups

Let ${L_0^\infty(G)}$ be the Banach space of all essentially bounded measurable functions on a locally compact group G vanishing at infinity. In this paper, we describe maximal and minimal left ideals in the Banach algebra ${L_0^\infty(G)^*}$ equipped with an Arens type product. We also investigate some spectral properties of ${L_0^\infty(G)^*}$ .     

Hyper MV ‐ideals in hyper MV ‐algebras

In this paper we define the hyper operations ?, ∨ and ∧ on a hyper MV ‐algebra and we obtain some related results. After that by considering the notions ofhyper MV ‐ideals and weak hyper MV ‐ideals, we prove some theorems. Then we determine relationships between (weak) hyper MV ‐ideals in a hyper MV ‐algebra (M, ⊕, *, 0) and (weak) hyper K ‐ideals in a hyper K ‐algebra (M, °, 0). Finally we give a characterization of hyper MV ‐algebras of order 3 or 4 based on the (weak) hyper MV ‐ideals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)