Atomistic subsemirings of the lattice of subspaces of an algebra

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.Comment: 14 page

The LV-hyperstructures

The largest class of hyperstructures is the one which satisfy the weak properties and they are called H v -structures introduced in 1990. The H v(c)-structures have a partial order (poset) on which gradations can be defined. We introduce the LV-construction based on the Levels Variable

Lie-Santilli admissible hyper-structures, from numbers to Hv-numbers

The class of Hv-structures defined on a set is very big and admits a partial order. For this reason, it has a numerous of applications in mathematics and other sciences as physics, biology, linguistics, to mention but a few. Here, we focus on the Lie-Santilli’s admissible case, where the hyper-numbers, called Hv-numbers, are used. In order to verify all needed axioms for Lie-Santilli’s admissibility, as the irreversibility and uniqueness of living organisms and time, on the one side and small results on the other side, we use the verythin Hv-fields. Therefore, we take rings and we enlarge only one result by adding only one element in order to obtain an Hv-field. This means that, we use only the associativity on the product and we transfer this to the weak-associativity on the hyper-product. Thus, from a semigroup on the product, we construct an Hv-group on the hyper-product