It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1, 2, . . . , n without repetition. These labellings are called Sn EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a n...

In this paper we extend some aspects of the theory of 'supersolvable lattices' [3] to a more general class of finite lattices which includes the upper-semimodular lattices. In particular, all conjectures made in [33 concerning upper-semimodular lattices will be proved. For instance, we will prove that if L is finite upper-semimodular and if L' denotes L with any set of 'levels' removed, then th...

A X g. We exhibit natural embeddings of M(A X) in M(A) that give rise to monomorphisms from 1 (M(A X)) to 1 (M(A)). We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of 1 (M(A)). Recall that X in L(A) is modular if X+Y is an element of L(A) for all Y in L(A). We call X in L(A) supersolvable if there exists a chain ...

A ring $A$ is (principally) nilary, denoted (pr-)nilary, if whenever $XY=0,$ then there exists a positive integer $n$ such that either $X^n=0$ or $Y^n=0$ for all (principal) ideals $X$, $Y$ of $A$. We determine necessary and/or sufficient conditions the group $A[G]$ to be nilary in terms on $G$. For example, we show that: (1) If (pr-)nilary and $G$ prime order each finite nontrivial normal subg...

We show that Terao's Conjecture ("Freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid") holds over fields with at most four elements. However, an example demonstrates that the field characteristic has to be fixed for this. 1. Free arrangements The present study continues an investigation of the connection between algebraic and combinator...

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, abelian groups, achieves same asymptotic worst-case query complexity as optimal randomized namely~$ \Order(\sqrt{ n}\, )$, where~$n$ is order of group. analogous algorithm non-abelian groups comes within a~$\sqrt{ \log n}$ factor complexity. best known Problem has \emph{expected\/} that sensitive to input,...