It is my privilege to introduce to this Society a vigorous and promising younger brother of group theory, by name, lattice theory. Among other things, I shall try to bring out the family resemblance. It is generally recognized that some familiarity with the notions of group, subgroup, normal subgroup, inner automorphism, commutator, and their technical properties, in a word, with group theory, ...

Abstract It is known that if G a group such the centre factor $$G/\zeta (G)$$ G / ζ ( ) polycyclic, then also commutator subgroup $$G'$$ ′ polycyclic. The aim of this paper to des...

Abstract An E -fuzzy group is a lattice-valued algebraic structure, defined on a crisp algebra which is not necessarily a group. The crisp equality is replaced by a particular fuzzy one denoted by E . Classical group-like properties are formulated as appropriate fuzzy identities special lattice theoretic formulas. We prove basic features of E -fuzzy groups: properties of the unit and inverses, ...

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a latti...

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not...

Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite group, example a subgroup of compact Lie group. We show that the electric Hamiltonian admits an interpretation as certain natural, nonunique Laplacian operator on Abelian or non-Abelian group and derive some consequences from this fact. Independent chosen Hamiltonian, provide full explici...

Sufficient conditions for the convergence in distribution of an infinite convolution product μ1 ∗μ2 ∗ . . . of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a refinement operator constructed from a measure μ and a dilation automorphism A. The existence of A implie...

The adjacency matrix of a symplectic dual polar graph restricted to the eigenspaces an abelian automorphism subgroup is shown act as weighted subspace lattice. connection between latter and Uq(sl2) used find irreducible components standard module Terwilliger algebra graphs. multiplicities isomorphic submodules are given.

Suucient conditions for the convergence in distribution of an inn-nite convolution product 1 2 : : : of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy T = where T is a reenement operator constructed from a measure and a dilation automorphism A. The existence of A implies G is nilpoten...