20 research outputs found
Morita Equivalence of C^*-Crossed Products by Inverse Semigroup Actions and Partial Actions
Morita equivalence of twisted inverse semigroup actions and discrete twisted
partial actions are introduced. Morita equivalent actions have Morita
equivalent crossed products
C*-actions of r-discrete groupoids and inverse semigroups
Groupoid actions on C*-bundles and inverse semigroup actions on C*-algebras
are closely related when the groupoid is r-discrete.Comment: LaTeX-2e, 18 pages, uses pb-diagram.st
Biased Weak Polyform Achievement Games
In a biased weak polyform achievement game, the maker and the breaker
alternately mark previously unmarked cells on an infinite board,
respectively. The maker's goal is to mark a set of cells congruent to a
polyform. The breaker tries to prevent the maker from achieving this goal. A
winning maker strategy for the game can be built from winning
strategies for games involving fewer marks for the maker and the breaker. A new
type of breaker strategy called the priority strategy is introduced. The
winners are determined for all pairs for polyiamonds and polyominoes up
to size four
Impartial achievement and avoidance games for generating finite groups
We study two impartial games introduced by Anderson and Harary and further
developed by Barnes. Both games are played by two players who alternately
select previously unselected elements of a finite group. The first player who
builds a generating set from the jointly selected elements wins the first game.
The first player who cannot select an element without building a generating set
loses the second game. After the development of some general results, we
determine the nim-numbers of these games for abelian and dihedral groups. We
also present some conjectures based on computer calculations. Our main
computational and theoretical tool is the structure diagram of a game, which is
a type of identification digraph of the game digraph that is compatible with
the nim-numbers of the positions. Structure diagrams also provide simple yet
intuitive visualizations of these games that capture the complexity of the
positions.Comment: 28 pages, 44 figures. Revised in response to comments from refere
Impartial avoidance games for generating finite groups
We study an impartial avoidance game introduced by Anderson and Harary. The
game is played by two players who alternately select previously unselected
elements of a finite group. The first player who cannot select an element
without making the set of jointly-selected elements into a generating set for
the group loses the game. We develop criteria on the maximal subgroups that
determine the nim-numbers of these games and use our criteria to study our game
for several families of groups, including nilpotent, sporadic, and symmetric
groups.Comment: 14 pages, 4 figures. Revised in response to comments from refere