Editing APS relation

{{basicapsdef}}

==Definition==

Let <math>(G,\\Phi)</math> be an [[APS]] of sets (possibly with additional structure). Then, a relation on <math>(G,\\Phi)</math>, defines, for every <math>n</math>, a relation <math>R_n</math> on <math>G_n</math> such that if <math>a, b</math> in <math>G_m</math> are related via <math>R_m</math> and <math>c,d</math> in <math>G_n</math> are related by <math>R_n</math>, then <math>\\Phi_{m,n}(a,c)</math> and <math>\\Phi_{m,n}(b,d)</math> in <math>G_{m+n}</math> are related via <math>R_{m+n}</math>

==Other notions==

===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>.

===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.