APS relation

From Apstheory

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Revision as of 23:30, 25 January 2007

Template:Basicdef

Contents 1 Definition 2 Other notions 2.1 Quotient map 2.2 Congruence

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 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.