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Definition with symbols
Given two subgroup properties and , the composition operator applied to these properties, denoted as , is the property defined as follows: has the property as a subgroup of if there is an intermediate subgroup such that satisfies as a subgroup of and satisfies as a subgroup of .
Given subgroup properties , , and , the following relation holds:
In other words, the composition operator is associative.
The composition operator is a monotone operator in both arguments, when the properties are given the usual partial order of implication. Further, it distributes over logical disjunction, and is hence a quantalic property operator.
The identity element for the composition operator is the property of being the improper subgroup, that is, of being the group embedded as a subgroup in itself. Any property that is implied by this property is termed an identity-true subgroup property.
The nil element for the composition operator is the fallacy subgroup property, that is the subgroup property that is never satisfied.
Since the composition operator is an associative quantalic property operator, the Transiter master theorem is applicable to it, and we can talk of left transiters, right transiters. We can also talk of the subordination with respect to this operator.