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 k1Ais 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. © 2013 Taylor and Francis Group, LLC.
Publication Source (Journal or Book title)
Communications in Algebra
Sage, D. (2013). Atomistic Subsemirings of the Lattice of Subspaces of an Algebra. Communications in Algebra, 41 (10), 3652-3667. https://doi.org/10.1080/00927872.2012.674588