Suppose <? denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) Res (£?), the class of all lattice-ordered groups which are subdirect products of groups in C; (2) Hyp(C), the class of lattice-ordered groups in Res(C) having all their ¿-homomorphic images in Res(<?); Para(C), the class of lattice-ordered gro...

We study the lattice of finite-index extensions of a given finitely generated subgroup H of a free group F . This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H . This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exp...

Remark. Apropos of reduction mod p: If V is a Qp-vector space and G ⊂ GL(V ) is a compact subgroup, then there exists a G-fixed lattice in V for the following reason. Pick any lattice L ⊂ V . Then the G-stabilizer of L is open and of finite index. So Λ = ∑ g∈G gL ⊂ V is also a lattice, and it is definitely G-stable. The same works with coefficients in any finite extension of Qp, or even in Qp (...

We study the lattice of finite-index extensions of a given finitely generated subgroup H of a free group F . This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H . This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exp...

We examine the relation between twisted versions of the extended supersymmetric gauge theories and supersymmetric orbifold lattices. In particular, for the N = 4 SYM in d = 4, we show that the continuum limit of orbifold lattice reproduces the twist introduced by Marcus, and the examples at lower dimensions are usually Blau-Thompson type. The orbifold lattice point group symmetry is a subgroup ...

Let G be a rank 2 complete affine or hyperbolic Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let P − 1 = B − t Bw1B be the negative standar...

Many studies have investigated the lattice structure of fuzzy substructures of algebraic sets such as group and ring. Some important results about modularity and distributivity have been obtained in these studies. In this paper, we first define the notion of (normal) (λ,μ)-L-subgroups on a group and investigate some of their properties. In particular, we discuss the relationships among ordinary...

In a previous paper, the second author established that, given finite fields F < E and certain subgroups C ≤ E×, there is a Galois connection between the intermediate field lattice {L | F ≤ L ≤ E} and C’s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product C⋊Gal(E/F ). However, the analysis when |F | i...

Freese and J6nsson [8] showed that the congruence lattice of a (universal) algebra in a congruence modular variety is always arguesian. On the other hand J6nsson [16] constructed arguesian lattices which cannot be embedded into the normal subgroup lattice of a group. These lattices consist of two arguesian planes of different prime order glued together over a two element sublattice (cf. Dilwort...

We describe a model for the interaction of the internal (spin) degree of freedom of a quantum lattice-gas particle with an environmental bath. We impose the constraints that the particle-bath interaction be fixed, while the state of the bath is random, and that the effect of the particlebath interaction be parity invariant. The condition of parity invariance defines a subgroup of the unitary gr...