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, 15:36, 28 April 2012→6.9. Higher Order Sign Relations : Introduction: markup
The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another that makes it difficult to know where to get on and where to get off the circle of explanation. As I currently understand their relationship, it can be approached in the following order:
The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another that makes it difficult to know where to get on and where to get off the circle of explanation. As I currently understand their relationship, it can be approached in the following order:
: '''''Relations have types.'''''
: '''''Types are functions.'''''
: '''''Functions are relations.'''''
In this setting, a ''type'' is a function from the ''places'' of a relation, that is, from the index set of its components, to a collection of sets that are called the ''domains'' of the relation.
In this setting, a ''type'' is a function from the ''places'' of a relation, that is, from the index set of its components, to a collection of sets that are called the ''domains'' of the relation.
A ''homogeneous relation'' is a relation, all of whose elementary relations have the same type. In this case, the type and the arity are properties that are defined for the relation itself. The rest of this discussion is specialized to homogeneous relations.
A ''homogeneous relation'' is a relation, all of whose elementary relations have the same type. In this case, the type and the arity are properties that are defined for the relation itself. The rest of this discussion is specialized to homogeneous relations.
When the arity of a relation is a finite number <math>k,\!</math> then the relation is called a <math>k\!</math>-place relation. In this case, the elementary relations are just the <math>k\!</math>-tuples belonging to the relation. In the finite case, for example, a ''non-trivial properly partial transaction'' is a <math>j\!</math>-tuple extracted from a <math>k\!</math>-tuple of the relation, where <math>1 < j < k.\!</math> The first element of an elementary relation is called its ''object'' or ''relate'', while the remaining elements are called its ''correlates''.
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1. Signs that denote single correlates of an object in a relation are called "higher ascent" (HA) signs.
1. Signs that denote single correlates of an object in a relation are called "higher ascent" (HA) signs.
