1 Abbreviations and Notations 2

The subject of this paper are general properties of direct sum decompositions of automata. Using certain properties of the lattice Sub(A) of subautomata of an automaton A and its Boolean part, lattices of direct sum congruences and direct sum decompositions of A are characterized. We show that every automaton A can be represented as a direct sum of direct sum indecomposable automata, and that t...

In 1988, Karchmer and Wigderson generalized Yao’s two-party communication model of functions to relations and showed a remarkable connection of this model to the Boolean circuit model. A few years later, continuing this line of work, Karchmer, Raz, and Wigderson proposed a program to separate NC from P through direct-sum-type inequalities in communication complexity. This spurred the study of t...

We study the connection between spectral properties of direct the sum of operators in the direct sum of Hilbert spaces and its coordinate operators.