#### Document Type

Article

#### Publication Date

10-1-2013

#### Abstract

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

#### First Page

3652

#### Last Page

3667

#### Recommended Citation

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