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We study the integral quaternions and the integral octonions along the combinatorics of the -cell, a uniform polytope with the symmetry , and the Gosset polytope with the symmetry . We identify the set of the unit integral octonions or quaternions as a Gosset polytope or a -cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the or actions on the or the -cell, respectively. Moreover, we show that each...
A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in with respect to a slightly different order and prove that this poset is...
In this paper we apply the notion of cell of a lattice for dealing with graph automorphisms of lattices (in connection with a problem proposed by G. Birkhoff).
In the present paper we generalize a result of a theorem of J. Jakubík concerning graph automorphisms of lattices to the case of multilattices of locally finite length.
This paper deals with the relations between graph automorphisms and direct factors of a semimodular lattice of locally finite length.
In it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later () this result was re-proved (and further improved on) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained...
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