we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

We prove that in the category of achimedean lattice-ordered groups with weak unit there is no homomorphism-closed monoreflection strictly between the strongest essential monoreflection (the so-called “closure under countable composition”) and the strongest monoreflection (the epicompletion). It follows that in the category of regular σframes, the only non-trivial monoreflective subcategory that...

We study the performance of finite frames for the encoding of vectors by applying first-order sigma-delta quantization to the frame coefficients. Our discussion is restricted to families of uniform tight frames, given by N vectors in d-dimensional Hilbert spaces, and mostly concerns the use of quantizers that assume only integer multiples of a step-size δ (mid-tread). We prove upper and lower b...

We construct a completely regular ordered space (X, T ,≤) such that X is an I-space, the topology T of X is metrizable and the bitopological space (X, T ♯,T ♭) is pairwise regular, but not pairwise completely regular. (Here T ♯ denotes the upper topology and T ♭ the lower topology of X.)

The free product CR S i of an arbitrary family of disjoint completely simple semigroups fS i g i2I , within the variety CR of completely regular semigroups, is described by means of a theorem generalizing that of Ka dourek and Poll ak for free completely regular semigroups. A notable consequence of the description is that all maximal subgroups of CR S i are free, except for those in the factors...

In this paper from q-ary perfect codes new completely regular q-ary codes are constructed. In particular, two new ternary completely regular codes are obtained from ternary Golay [11, 6, 5] code. The first [11, 5, 6] code with covering radius ρ = 4 coincides with the dual Golay code and its intersection array is (22, 20, 18, 2, 1; 1, 2, 9, 20, 22) . The second [10, 5, 5] code, with covering rad...

Completely regular semigroups CR are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by L.CR/. Let B denote the variety of bands and L.B/ the lattice of its subvarieties. The mapping V → V ∩ B is a complete homomorphism of L.CR/ onto L.B/. The congruence induced by it has classes that are intervals, say VB = [VB ;V B] for...