A. Sakowicz

2003

Citations

1

Influential Citations

2

Citations

Journal

Colloquium Mathematicum

Abstract

We give the description of locally finite groups with strongly balanced subgroup lattices and we prove that the strong uniform dimension of such groups exists. Moreover we show how to determine this dimension. 1. Lattice preliminaries. All lattices considered in this paper have the least and the greatest element, denoted by 0 and 1 respectively. They do not need to be finite (in contrast to [1, 9]). We will apply also some other notation and terminology about lattices, as in [2, 9]. In particular if L is a lattice we will say that L is balanced if for all x, y, z ∈ L we have x ∧ y = 0 & (x ∨ y) ∧ z = 0 ⇒ (y ∨ z) ∧ x = 0 & (z ∨ x) ∧ y = 0, and L is strongly balanced if all nonempty intervals of L are balanced. It is easy to show that every distributive and even modular lattice is strongly balanced. Hence strong balancedness can be considered as a generalization of modularity. In [5], with motivation coming from Theorem 6.1.10 of [12], nearly modular lattices were introduced, also as a generalization of modular ones. In the last section of this paper we indicate that there is no inclusion between the class of strongly balanced and the class of nearly modular lattices. Further properties of balanced and strongly balanced lattices can be found in [9, 10, 13]. If a, u ∈ L then, as in [6, 10], we will say that a is essential in L if a ∧ x 6= 0 for every 0 6= x ∈ L, and u is uniform in L if u 6= 0 and every element from (0, u] is essential in [0, u]. For example any atom is a uniform element and 1 is an essential element in every nontrivial lattice. Let L be a lattice. It will be called locally uniform ([10]) if any nontrivial interval [0, a] ⊆ L contains a uniform element, and strongly locally uniform if any of its nontrivial intervals is locally uniform. Clearly every (strongly) atomic lattice is (strongly) locally uniform. Notably, any finite lattice is 2000 Mathematics Subject Classification: 20E15, 20E25.

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