Forbidden configurations for distributive and modular ordered sets

As modular and distributive ordered sets generalize modular and distributive lattices, it is a natural question to ask whether there exist some forbidden configurations for those ordered sets. We present such configurations in the form of strong subsets and LU subsets.     

The automorphism group of the universa distributive lattice

We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice and of the countable atomless generalized Boolean lattice. Also, we show that subgroups of these automorphism groups of index less than lie between the pointwise and the setwise stabilizer of a finite set. Received March 8, 1999; accepted in final form September 16, 1999.     

A theorem of ordered duality

We associate, to any ordered configuration of n points or ordered arrangement of n lines in the plane, a periodic sequence of permutations of [1, n] in a way which reflects the order and convexity properties of the configuration or arrangement, and prove that a sequence of permutations of [1, n] is associated to some configuration of points if and only if it is associated to some arrangement of lines. We show that this theorem generalizes the standard duality principle for the projective plane, and we use it to derive duals of several well-known theorems about arrangements, including a version of Helly's theorem for convex polygons.     

分配格中模糊理想诱导的模糊幂格

在模糊幂格讨论基础上,给出了由分配格中模糊理想诱导的两种模糊幂格.     

Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattice

In this paper, the concept of lattice over generalized intuitionistic fuzzy matrices (GIFMs) are introduced and have shown that the set of GIFMs forms a distributive lattice. Some algebraic properties of generalized intuitionistic fuzzy matrices (GIFMs) are presented over distributive lattice. Also, some characteristics of generalized intuitionistic fuzzy nilpotent matrices (GIFNMs) are discussed over distributive lattice. Finally, the reduction of GIFNMs over distributive lattice are given with some properties.     

Remarks on Priestley duality for distributive lattices

The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix.     

对称扩展的有界分配格的同余关系及其应用

给出了对称扩展的有界分配格的定义,即带有满足一定条件的一元运算的有界分配格.然后给出了这种分配格上的主同余的等式刻划及其可补性.最后,讨论了对称扩展的有界分配格的次直不可约性。     

A topology on lattice ordered groups

We show that a lattice ordered group can be topologized in a natural way. The topology depends on the choice of a set of admissible elements (-topology). If a lattice ordered group is 2-divisible and satisfies a version of Archimedes' axiom (-group), then we show that the -topology is Hausdorff. Moreover, we show that a -group with the -topology is a topological group.

     

Affine completeness and wreath product decompositions of lattice ordered group

Let Δ and H be a nonzero abelian linearly ordered group or a nonzero abelian lattice ordered group, respectively. In this paper we prove that the wreath product of Δ and H fails to be affine complete. Supported by VEGA grant 2/4134/24. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, grant I/2/2005.     

A compactly generated lattice with unique complements is distributive

It is known that in a lattice with unique complements, the modularity, atomicity, de Morgan, and other conditions imply that the lattice is distributive. It is proved that a compactly generated lattice with unique complements is also Boolean.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 617–620, November, 1972.     

On a relative uniform completion of an archimedean lattice ordered group

The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups. Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) of G is dealt with. It is known that exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in . Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group

In this paper, we investigate the properties of normal fuzzy subgroups of a fuzzy group. We also introduce the notion of a characteristic fuzzy subgroup of a fuzzy group, of which we provide level subset and strong level subset characterizations. Then we prove that a characteristic fuzzy subgroup of a fuzzy group is a normal fuzzy subgroup. Besides we prove that the commutator subgroup generated by the commutator of two normal fuzzy subgroups of a fuzzy group is contained in their intersection. Finally, we construct many new lattices and sublattices of fuzzy subgroups and normal fuzzy subgroups of a given fuzzy group. We also construct lattices of characteristic fuzzy subgroups possessing sup-property.     

A note on uncountable groups with modular subgroup lattice

In this paper, we obtain a rigidity theorem by modifying Cheng–Yau’s technique to linear Weingarten submanifolds in the unit sphere S^n+p(1) with parallel normalized mean curvature vector. As a corollary, we have Theorem 1.3 in Guo and Li (Tohoku Math J 65:331–339, 2013) and Theorem 2 in Li (Math Ann 305:665–672, 1996).