Modular lattices over cyclotomic fields

This paper is concerned with modular lattices over cyclotomic fields. In particular, the notion of Arakelov modular ideal lattice is introduced. All the cyclotomic fields over which there exists an Arakelov modular lattice of given level are characterised.     

Partitions with initial repetitions

A variety of interesting connections with modular forms, mock theta functions and Rogers- Ramanujan type identities arise in consideration of partitions in which the smaller integers are repeated as summands more often than the larger summands. In particular, this concept leads to new interpretations of the Rogers Selberg identities and Bailey＇s modulus 9 identities.     

On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements

Gel’fand and Ponomarev [11] introduced the concept of perfect elements and constructed such in the free modular lattice on 4 generators. We present an alternative construction of such elements u (linearly equivalent to theirs) and for each u a direct decomposition u, [`(u)]{\bar{u}} of the generating quadruple within the free complemented modular lattice on 4 generators; u, [`(u)]{\bar{u}} are said to form a perfect pair. This builds on [17] and fills a gap left there. We also discuss various notions of perfect elements and relate them to preprojective and preinjective representations.     

幂格的素理想和素对偶理想

在文 [1 ]中 ,引入了幂格的概念 ,并讨论了其相关性质 .本文在此基础上 ,讨论格与其幂格的理想 ,对偶理想的关系 ,以及格与其幂络的素理想 ,素对偶理想的关系 .     

Decomposition of ideals in Abelianp-extensions of complete, discretely valuated fields

A decomposition of an ideal as a Galois module in an Abelian p-extension of a complete, discretely valuated field into indecomposable summands is found. Bibliography: 5titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 23–33.     

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ?-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ?-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ?-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.     

New Light on Bergman Complexes by Decomposing Matroid Types

We give a shorter proof of the fact, that Bergman complexes of matroids can be subdivided to realizations of the nested set complexes of the lattice of flats. Then, we present a direct sum decomposition into connected summands of the matroid types of faces of Bergman complexes.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Über Morphismen halbmodularer Verbände

Morphisms and weak morphisms extend the concept of strong maps and maps of combinatorial geometry to the class of finite dimensional semimodular lattices. Each lattice which is the image of a semimodular lattice under a morphism is semimodular. In particular, each finite lattice is semimodular if and only if it is the image of a finite distributive lattice under a morphism. Regular and non-singular weak morphisms may be used to characterize modular and distributive lattices. Each morphism gives rise to a geometric closure operator which in turn determines a quotient of a semimodular lattice. A special quotient, the Higgs lift, is constructed and used to show that each morphism decomposes into elementary morphisms, and that each morphism may be factored into an injection and a contraction.

     

A study on distributive and modular lattice ordered fuzzy soft group and its duality

Soft set theory has a rich potential application in several fields. A soft group is a parameterized family of subgroups and a fuzzy soft group is a parameterized family of fuzzy subgroups. The concept of fuzzy soft group is the generalization of soft group. Abdulkadir Aygunoglu and Halis Aygun introduced the notion of fuzzy soft groups in 2009[1]. In this paper, the concept of lattice ordered fuzzy soft groups and its duality has been introduced. Then distributive and modular lattice ordered fuzzy soft groups are analysed. The objective of this paper is to study the lattice theory over the collection of fuzzy soft group in a parametric manner. Some pertinent properties have been analysed and hence established duality principle.