Matrix IAPS
From Apstheory
Current revision as of 03:11, 30 December 2006
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[edit] Definition
[edit] Symbol-free definition
The matrix IAPS over a base ring is defined as the IAPS whose members are the matrix rings over the base ring and whose block concatenation maps are the usual block concatenation maps of matrices.
It is an IAPS of algebras over the base ring.
[edit] Definition with symbols
The matrix IAPS over a base ring Failed to parse (Can't write to or create math temp directory): R
is denoted as Failed to parse (Can't write to or create math temp directory): Mat(R)
, and is defined as follows:
- The Failed to parse (Can't write to or create math temp directory): n^{th}
member is the matrix ring Failed to parse (Can't write to or create math temp directory): Mat_n(R) of Failed to parse (Can't write to or create math temp directory): n × Failed to parse (Can't write to or create math temp directory): n matrices over Failed to parse (Can't write to or create math temp directory): R with usual matrix addition and multiplication.
- The block concatenation of a matrix of order Failed to parse (Can't write to or create math temp directory): m
with a matrix of order Failed to parse (Can't write to or create math temp directory): n is a matrix of order Failed to parse (Can't write to or create math temp directory): m+n whose top left corner is the first matrix, and bottom right corner is the second matrix, with the remaining entries being zeroes.
[edit] Subs of the matrix IAPS=
[edit] Monoid-theoretic subs (multiplicative)
With respect to the multiplicative structure ,the matrix IAPS forms an IAPS of monoids. A natural sub-IAPS is the sub-IAPS comprising invertible matrices. This is an IAPS of groups and is termed the GL IAPS. The GL IAPS is very important from the viewpoint of group IAPSes and has a number of important subs.
The power APS over the base ring is also a sub-IAPS of the matrix IAPS.
