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Regular parallelisms in kinematic spaces
No full textHere we propose a definition of regular parallelism in a linear space not necessarily embedded onto a projective space and we investigate its properties in the particular case of kinematic spaces. We prove that the kinematic parallelisms are always regular in that sense and we deduce some results on the group of translations acting transitively on the pointset
Translation structures with a principal line
Full text linkEnHere we study affine parallel translation structures, both finite and infinite, with a principal line, that is a line which intersects every line not in its parallel class. These structures can be regarded also as (finite or infinite) translation transversal divisible designs. An algebraic characterization of these structures in terms of semidirect product of groups is provided and the main properties related to their group of automorphisms are inspected. The particular case of kinematic spaces is also taken into consideration
Slid product of loops: a generalization
No full textIn this paper, following the slid product construction for loops with inverses of Pasotti and Zizioli [J Geom 100(1–2):129–145, 2011], we present the general setting in order to build up a new loop (L, ⊕) starting from loops (K, +) equipped with a well ordering “<=”, (P, +) and (P, +) with the same neutral element. The results established in the aforementioned note are generalized as well. Moreover we investigate the nuclei of L, the normality of subloops isomorphic to (K, +) and (P, +) and discuss some examples
A construction of loops by means of regular permutation sets
No full textFollowing the slid product construction for loops with inverses of [Pasotti, S. and E. Zizioli, Loops with two-sided inverses constructed by a class of regular permutation sets, J. Geom. 100 (2011), pp. 129–145. URL http://dx.doi.org/10.1007/s00022-011-0071-5], we present here the general setting ensuring the construction of a new loop (L,+) starting from loops (K,+), (P,+') and (P,+) possessing suitable properties. The results established in the aforementioned note are generalized as well. Moreover we investigate the nuclei of L and the normality of subloops isomorphic to (K,+) and (P,+')
Corrigendum to The limit rotation loop of a hyperbolic plane [Applied Mathematical Sciences, 7, 117-120, (2013), 5863-5878]
No full textThere is a small gap in the proof of Theorem 4.5 of the mentioned paper. Here the authors give the complete proof
Loops, regular permutation sets and colourings of directed graphs
No full textWe establish a correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring and characterize regular permutation sets and, respectively, colored graphs giving rise to the same loop, to isomorphic loops and to isotopic loops
Loops, regular permutation sets and graph colourings
No full textWe establish a correspondence among loops, regular permutation sets and directed graphs with a suitable edge colouring and relate some algebraic properties of the loop to configurations of the associated graph
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