Opened 4 years ago

Closed 4 years ago

## #845 closed proposed feature (invalid)

# mereotopology of solid physics

Reported by: | anonymous | Owned by: | somebody |
---|---|---|---|

Priority: | critical | Milestone: | 3-1-5 and higher |

Component: | dontKnow | Version: | 3-1-7 |

Keywords: | mereotopology | Cc: | su.azadeh.2005@… |

### Description

Let A be a ring, G a finite group acting on /i and y: G X G -) C&4), the
units of/i, a map satisfying
(1) y(g, 6) Hgg’, 6’) = g(W, 6’)) y(g, g’g”) for g, g’, d’ in G
(2) y(e, g) = 1 = y(g, e) for g E G, e the identity element of G,
(3) y(g, g’)kg’W) = &f(n)) y(g, g’> for g, g’ in G.
Then the corresponding crossed product algebra /i * ?G, or /i x G for
short, has an elements CgiEG A.-; ,gi li ~/i. Addition is componentwise, and
multiplication is given by & = g(A)g and g, g2 = y(g, , g2) m. In this
paper we assume that the values of y lie in the center Z(A) of/i. Hence the
action of G on /i is given by a group homomorphism G-P Aut(/i), and (3)
can be left out. In the special case that y is the trivial map we write /1G
instead of/i * G, and the elements as C_{,c li gi. AG is then called a skew
group ring.
There is a lot of literature on skew group algebras and crossed product
algebras, and on the relationship with the ring AC whose elements are those
elements of A left fixed by G. Much work has been done on which properties
of li are inherited by n * G or AC. Some of the work on the relationship
between these rings has its roots in trying to develop a Galois theory for
noncommutative rings. We refer to [3, 7, 13-15, 19, 21, 23-25, 27, 281 and
their references.
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