Recent research has demonstrated that the strength of a

transitive relation characterized by a given nodal number is always greater than the equivalence relation of the same nodal number (i.e., the relational type effect; Doran & Fields, 2012; Fields, 2015).

In this paper, we investigate a new extension of the transitive closure concept: "the transitive closure according to a given property." That is, for a given property P, and a relation R, we are interested in computing the smallest

transitive relation containing R such that the property P holds.

A simple geometric argument (see Proposition 3.5 for details) shows that in a thin building parallelism is a

transitive relation and thus is an equivalence relation on the set of all the residues.

This expression [alpha](G, x) induces a quasi-order relation (i.e., reflexive and

transitive relation) on the set of all graphs with n vertices.

Let [??] be the set of implicit precedence rules automatically synthesized from the defeasible rules declared in [DELTA].Clearly, [I.sub.[DELTA]] is an irreflexive relation (since the specificity criterion is defined only for conflicting rules), [I.sub.[DELTA]] is an asymmetric relation (since, if [??], the antecedent of l' is defeasibly derived from the antecedent of l, but not vice-versa), and [I.sub.[DELTA]] is a

transitive relation, with respect to conflicting rules (since, if [??] and [??], then [??] and the antecedent of [l.sup.m] is defeasibly derivable from the antecedent of l, but not vice-versa).

It happens that this relation comes out to be a

transitive relation (allowing for ranking of solutions, as well as fast search algorithms) and can be used to valuate the efficiency losses related to making fair allocations.

While logical implication based on classification is a

transitive relation, causation in terms of dispositions is not.

If we take an arbitrary relation R on M then we can consider the smallest

transitive relation containing R, so-called transitive closure, namely [R.sup.*] = [[union].sub.i[member of]N] [R.sup.i] where [R.sup.i] is the composition of i copies of R.

i) [bar.Q] is the smallest

transitive relation containing Q

We first prove ~[sub.I] is a reflexive, symmetric and

transitive relation, i.e., it is an equivalence relation.

When analyzing a pattern, the child can either make use of the

transitive relation or, since all of the elements in the sequence ABC are present simultaneously, compare A and C directly, even though B intervenes.

Furthermore, most (if not all) give special names to relations with other combinations of properties--e.g., a partial ordering is a reflexive, antisymmetric, and

transitive relation, and a total ordering is a partial ordering with the additional property that any two elements of the universe of discourse are related either by the ordering or by its converse.