some theorems on strict betweenness relations

Let $B$ be a strict betweenness relation. In the following the sets $B_{*pq},B_{p*q},B_{pq*},B_{pq},B(p,q)$ are defined in the entry about some theorems on the axioms of order.

Theorem 1.

Three elements arein a strict betweenness relation only if they are pairwise distinct.

Theorem 2.

If $B$ is strict, then $B_{*pq}$ , $B_{p*q}$ and $B_{pq*}$ are pairwise disjoint.Furthermore, if $p=q$ then all three sets are empty.

Theorem 3.

If $B$ is strict, then $B_{pq}\\cap B_{qp}=B_{p*q}$ and $B_{pq}\\cup B_{qp}=B(p,q)$ .

Theorem 4.

If $B$ is strict, then for any $p,q\\in A$ , $p\eq q$ , $B_{*pq}$ , $B_{p*q}$ and $B_{pq*}$ are infinite.

单词	SomeTheoremsOnStrictBetweennessRelations
释义	some theorems on strict betweenness relations Let $B$ be a strict betweenness relation. In the following the sets $B_{pq},B_{pq},B_{pq},B_{pq},B(p,q)$ are defined in the entry about some theorems on the axioms of order. Theorem 1. Three elements arein a strict betweenness relation only if they are pairwise distinct. Theorem 2. If $B$ is strict, then $B_{pq}$ , $B_{pq}$ and $B_{pq}$ are pairwise disjoint.Furthermore, if $p=q$ then all three sets are empty. Theorem 3. If $B$ is strict, then $B_{pq}\\cap B_{qp}=B_{pq}$ and $B_{pq}\\cup B_{qp}=B(p,q)$ . Theorem 4. If $B$ is strict, then for any $p,q\\in A$ , $p\eq q$ , $B_{pq}$ , $B_{pq}$ and $B_{pq}$ are infinite.
随便看	Matroid matroid MatroidIndependenceAxioms matroid independence axioms MaximalBipartiteMatchingAlgorithm maximal bipartite matching algorithm MaximalCondition maximal condition MaximalElement maximal element MaximalErgodicTheorem Maximal ergodic theorem MaximalIdeal maximal ideal MaximalIdealIsPrime maximal ideal is prime MaximalIdealIsPrimegeneralCase maximal ideal is prime (general case) MaximalIdealsOfRingOfFormalPowerSeries maximal ideals of ring of formal power series MaximalIdealsOfTheAlgebraOfContinuousFunctionsOnACompactSet maximal ideals of the algebra of continuous functions on a compact set MaximalIntervalOfExistenceOfOrdinaryDifferentialEquations maximal interval of existence of ordinary differential equations MaximalityPrinciple