All Finite Distributive Lattices Occur as Intervals Between Hausdorff Topologies

It is shown that a finite lattice L is isomorphic to the interval betweentwo Hausdorff topologies on some set if and only if L is distributive. Thecorresponding results had previously been shown in ZFC for intervals between T₁ topologies and, assuming the existence of infinitely manymeasurable cardinals, for intervals between T₃ topologies.     

内射拓扑分子格

本文证明了任给T_O拓扑分子格(L,η),以下三条等价:(1)(L,η)为正则内射拓扑分子格;(2)L为完备集环且其完备余素元集ht(L)形成一连续格,余拓扑η为该连续格ht(L)上的Scott闭集格;(3)存在T_O内射拓扑空间(X,Τ),(L,η)同胚于(P(X),Τ~c)在拓扑分子格范畴中的Sober化。此外,还给出了正则内射拓扑分子格、(一般)内射拓扑分子格以及正则内射分子格的一般结构。作为应用,重新证明了有指数元的拓扑分子格的结构。     

A Minimax Theorem in Banach Lattices

This paper is mainly concerned with measures which take their values in a Banach lattice. Examples of capacities are given and a type of minimax theorem for vector capacities is proved.     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

Layer-Projective Lattices. II

The main result of this paper is: any primary Arguesian lattice over the field GF(p) of geometric dimension at least three is isomorphic to the lattice of all submodules of a finitely generated module over the ring of polynomials of bounded degree over the field GF(p).     

Lattices of subgroup and subsystem functors

Lattices of subgroup and subsystem functors are investigated. In particular, it is proved that for the case where X is a formation of finite groups and width of the lattice F₀(X) is at most |π(X)|, the formation X is metanilpotent and |π(X)| ⩽ 3. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 710–730, November–December, 2006.     

分配格中理想诱导的幂格

本文在文 [1 ]的基础上 ,给出由分配格中理想诱导的两种幂格 .     

Pseudocomplementation in (Normal) Subgroup Lattices

The goal of this article is to study finite groups admitting a pseudocomplemented subgroup lattice (PK-groups) or a pseudocomplemented normal subgroup lattice (PKN-groups). In particular, we obtain a complete classification of finite PK-groups and of finite nilpotent PKN-groups. We also study groups with a Stone normal subgroup lattice, and we classify finite groups for which every subgroup has a Stone normal subgroup lattice. Finally, we obtain a complete classification of finite groups for which every subgroup is monolithic.     

Martingales in Banach Lattices

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E_n) on a Banach lattice F is said to be a filtration if E_nE_m = E_{n∧ m}. A sequence (x_n) in F is a martingale if E_nx_m = x_n whenever n ≤ m. Denote by M = M(F, (E_n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn’t contain a copy of c₀ or if every E_n is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. Mathematics Subject Classification (2000). 60G48, 46B42     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

Sublattices of Lattices of Convex Subsets of Vector Spaces

Let Co(V) be a lattice of convex subsets of a vector space V over a totally ordered division ring . We state that every lattice L can be embedded into Co(V), for some space V over . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional; in this case L also embeds into a finite lower bounded lattice of the form Co(V,)={X | X Co(V)}, for some finite subset of V. This result yields, in particular, a new universal class of finite lower bounded lattices.     

Collection of Finite Lattices Generated by a Poset

It is proved that the collection of all finite lattices with the same partially ordered set of meet-irreducible elements can be ordered in a natural way so that the obtained poset is a lattice. Necessary and sufficient conditions under which this lattice is Boolean, distributive and modular are given.     

Lattices in the Pseudoeuclidean Plane

Because of the metric structure of the pseudoeuclidean plane, lattices have some remarkable features, e.g., concerning their symmetry groups and concerning isotropic, space- and time-like elements. Especially (p,q,s)-lattices, which are characterized by the fact that their symmetry groups contain hyperbolic rotations, are showing some interesting structures and substructures.     

Lattices embeddable in subsemigroup lattices. I. Semilattices

V. B. Repnitskii showed that any lattice embeds in some subsemilattice lattice. In his proof, use was made of a result by D. Bredikhin and B. Schein, stating that any lattice embeds in the suborder lattice of a suitable partial order. We present a direct proof of Repnitskii’s result, which is independent of Bredikhin—Schein’s, giving the answer to a question posed by L. N. Shevrin and A. J. Ovsyannikov. We also show that a finite lattice is lower bounded iff it is isomorphic to the lattice of subsemilattices of a finite semilattice that are closed under a distributive quasiorder. Supported by INTAS grant No. 03-51-4110; RF Ministry of Education grant No. E02-1.0-32; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1; a grant from the Russian Science Support Foundation; SB RAS Young Researchers Support project No. 11. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 215–230, March–April, 2006.     

Atoms of Lattices of Supernilpotent and Special Radicals

ＡｔｏｍｓｏｆＬａｔｔｉｃｅｓｏｆＳｕｐｅｒｎｉｌｐｏｔｅｎｔａｎｄＳｐｅｃｉａｌＲａｄｉｃａｌｓＺｈｏｎｇＤｅｑｉａｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＢａｓｉｃＣｏｕｒｓｅｓ，ＺｈｕｚｈｏｕＥｎｇｉｎｅｅｒｉｎｇＩｎｓｔｉｔｕｔｅ）Ａｂｓｔｒａｃｔ：Ｉ...     

Vector Lattices on a Set of Two Generators

It is proved that the center of an automorphism group Aut(FVL2) of a free vector lattice FVL2 on a set of two free generators is isomorphic to a multiplicative group of positive reals. It is shown that the free vector lattice FVL2 has an isomorphic representation by continuous piecewise linear functions of the real line; as a consequence, the ideal lattice and the root system for rectifying ideals in FVL2 are amply described. Similar results are obtained for a free vector lattice FVL2 _Q 2 generated by two elements over a field of rational numbers.     

分子格上伪补的构造

本文给出了分子格上伪补的构造定理，并证明了当一个非平凡分子格上有伪补，其既约因子上并不一定存在伪补，但其准既约因子上一定存在伪补，且任一分子格上的伪补都可表为其准既约因子上伪补的直积．     

完备格中的成分理论

在完备格中引入了元素的成分概念．基于此,引入了元素的宽度的概念．在分配格的情形证明了元素的成分集对有限并运算封闭且有某种遗传性．证明了元素的有限宽度的成分之集是定向集．称没有非平凡成分的元素为颗粒．称每个非零元素都可表示为其颗粒成分之并的完备格为颗粒表示格．证明了拓扑空间是局部连通的充要条件是其开集格为颗粒表示格．     

On Countable Products of Finite Hausdorff Spaces

We investigate in ZF (i.e., Zermelo‐Fraenke set theory without the axiom of choice) conditions that are necessary and sufficient for countable products ∏_m∈ℕX_m of (a) finite Hausdorff spaces X_m resp. (b) Hausdorff spaces X_m with at most n points to be compact resp. Baire. Typica results: (i) Countable products of finite Hausdorff spaces are compact (resp. Baire) if and only if countable products of non‐empty finite sets are non‐empty. (ii) Countable products of discrete spaces with at most n + 1 points are compact (resp. Baire) if and only if countable products of non‐empty sets with at most n points are non‐empty.     

Note on Reflexive Algebras with Non—distributive Invariant Lattices

§０．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＬｂｅａｃｏｌｌｅｃｔｉｏｎｏｆｃｌｏｓｅｄｓｕｂｓｐａｃｅｓｏｆａＨｉｌｂｅｒｔｓｐａｃｅＨｃｏｎｔａｉｎｉｎｇ｛０｝ａｎｄＨｗｈｉｃｈｆｏｒｍｓａｃｏｍｐｌｅｔｅｓｕｂｓｐａｃｅｌａｔｔｉｃｅｕｎｄｅｒｔｈｅｏｐｅｒａｔｉｏｎｓ∧（ｉｎｔｅｒｓｅｃｔｉｏｎ）ａｎｄ∨（ｃｌｏｓｅｄｌｉｎｅａｒｓｐａｎ）．ＷｅｄｅｎｏｔｅｂｙａｌｇＬｔｈｅａｌｇｅｂｒａｏｆａｌｌｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｓｗｈｉｃｈｌｅａｖｅａｌｌｓｕｂｓｐａｃｅｓｉｎＬｉ…