Baer [1] observed that modular lattices of finite length (for example subgroup lattices of abelian groups) can be conceived as subspace lattices of a projective geometry structure on an ordered point set; the set of join irreducibles which in this case are the cyclic subgroups of prime power order. That modular lattices of finite length can be recaptured from the order on the points and, in add...

After the research on naBL-algebras gained by non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of generalization BL-algebras, which can be obtained general functions. In this paper, we show that not every function induce an BL-algebra counterexample thus propose new concept weak BL-algebras. We prove each corresponds to BL-algebra; there...

For a distributive lattice L, we consider the problem of interpolating functions f : D → L defined on a finite set D ⊆ L, by means of lattice polynomial functions of L. Two instances of this problem have already been solved. In the case when L is a distributive lattice with least and greatest elements 0 and 1, Goodstein proved that a function f : {0, 1} → L can be interpolated by a lattice poly...

In this note we analyse the Lie algebras of physical states stemming from lattice constructions on general even, self-dual lattices Γ with p ≥ q. It is known that if the lattice is of rank ≤ 26 and at most Lorentzian, the resulting Lie algebra is of generalized Kac-Moody type (or has a quotient that is). We show that this is not true as soon as q > 1. By studying a certain sublattice in the cas...

In this article, we study the geometry of units and ideals cyclotomic rings derive an algorithm to find a mildly short vector in any given ideal lattice quantum polynomial time, under some plausible number-theoretic assumptions. More precisely, ring conductor m , finds approximation shortest by factor exp (Õ(√ )). This result exposes unexpected hardness gap between these structured lattices gen...

Let S be a set of n ideals of a commutative ring A and let Geven (respectively Godd) denote the product of all the sums of even (respectively odd) number of ideals of S. If n ≤ 6 the product of Geven and the intersection of all ideals of S is included in Godd. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.