APS relation
From Apstheory
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===Quotient map=== | ===Quotient map=== | ||
- | An APS relation defines a set-theoretic map to a set-theoretic quotient APS, where the <math>n^{th}</math> member is the collection of equivalence classes of <math>G_n</math> with respect to <math>R_n</math>. | + | An APS relation which is an equivalence relation at each member defines a set-theoretic map to a set-theoretic quotient APS, where the <math>n^{th}</math> member is the collection of equivalence classes of <math>G_n</math> with respect to <math>R_n</math>. |
===Congruence=== | ===Congruence=== | ||
An [[APS congruence]] is an APS relation if, for every <math>n</math>, the relation at <math>n</math> is a congruence. Note that any APS congruence must first of all be a set-theoretic equivalence relation. | An [[APS congruence]] is an APS relation if, for every <math>n</math>, the relation at <math>n</math> is a congruence. Note that any APS congruence must first of all be a set-theoretic equivalence relation. |
Revision as of 23:34, 25 January 2007
Contents |
Definition
Let Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
be an APS of sets (possibly with additional structure). Then, a relation on Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
, defines, for every Failed to parse (Can't write to or create math temp directory): n , a relation Failed to parse (Can't write to or create math temp directory): R_n
on Failed to parse (Can't write to or create math temp directory): G_n such that if Failed to parse (Can't write to or create math temp directory): a, b in Failed to parse (Can't write to or create math temp directory): G_m are related via Failed to parse (Can't write to or create math temp directory): R_m and Failed to parse (Can't write to or create math temp directory): c,d in Failed to parse (Can't write to or create math temp directory): G_n are related by Failed to parse (Can't write to or create math temp directory): R_n
, then Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}(a,c)
and Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}(b,d) in Failed to parse (Can't write to or create math temp directory): G_{m+n} are related via Failed to parse (Can't write to or create math temp directory): R_{m+n}
Other notions
Quotient map
An APS relation which is an equivalence relation at each member defines a set-theoretic map to a set-theoretic quotient APS, where the Failed to parse (Can't write to or create math temp directory): n^{th}
member is the collection of equivalence classes of Failed to parse (Can't write to or create math temp directory): G_n with respect to Failed to parse (Can't write to or create math temp directory): R_n
.
Congruence
An APS congruence is an APS relation if, for every Failed to parse (Can't write to or create math temp directory): n , the relation at Failed to parse (Can't write to or create math temp directory): n
is a congruence. Note that any APS congruence must first of all be a set-theoretic equivalence relation.