Sub-APS

From Apstheory

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Since the <math>\\Phi</math> is understood for the sub-APS, we may omit it and simply say that <math>H</math> is a sub-APS of <math>G</math>.
Since the <math>\\Phi</math> is understood for the sub-APS, we may omit it and simply say that <math>H</math> is a sub-APS of <math>G</math>.
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==Other notions==
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===Smallest sub-APS containing a given collection of subobjects===
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If we are given a subobject of <math>G_n</math> for every <math>n</math>, we can talk of the sub-APS generated by these subobjects. This associates to every <math>n</math>, the set of elements in <math>G_n</math> that arise via a block concatenation of elements upto <math>n</math>.
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===Intersection of sub-APSes===
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Given two sub-APSes <math>H</math> and <math>K</math> of an APS <math>G</math>, the intersection of <math>H</math> and <math>K</math> associates to each <math>n</math> the subset <math>L_n = H_n</math> &cap; <math>K_n</math> of <math>G_n</math>. If the intersection of two subobjects is a subobject (which, for instance, is the case in algebraic structures) then <math>L</math> is also a sub-APS of <math>G</math>.
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Similarly, if an arbitrary intersection of subobjects is a subobject, then an arbitrary intersection of sub-APSes is a sub-APS.
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Revision as of 08:10, 9 March 2012

This article gives a basic definition in the APS theory. It is strictly local to the wiki

Definition

Let Failed to parse (Can't write to or create math temp directory): (G,\\Phi) 

be an APS over a monoidal concrete category. Then a sub-APS of Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
associates to each Failed to parse (Can't write to or create math temp directory): n
a subobject Failed to parse (Can't write to or create math temp directory): H_n
of Failed to parse (Can't write to or create math temp directory): G_n
such that the restriction of Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
to Failed to parse (Can't write to or create math temp directory): H_m
× Failed to parse (Can't write to or create math temp directory): H_n
takes it inside Failed to parse (Can't write to or create math temp directory): H_{m+n}

.

Thus we can view Failed to parse (Can't write to or create math temp directory): (H,\\Phi) 

as an APS in its own right (note that since the associativity condition is satisfied for the block concatenation on Failed to parse (Can't write to or create math temp directory): G

, it is also satisfied for the block concatenation on Failed to parse (Can't write to or create math temp directory): H .

Since the Failed to parse (Can't write to or create math temp directory): \\Phi 

is understood for the sub-APS, we may omit it and simply say that Failed to parse (Can't write to or create math temp directory): H
is a sub-APS of Failed to parse (Can't write to or create math temp directory): G

.

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