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Further development of the theory of join systems
APPENDIX Examples of join systems
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abelian group aEb(mod Af\B apply arbitrary aséb associative law basic classical geometry CLOSURE LAW commutative laws congruence contains Conversely suppose convex set Corol Corollary deﬁne deﬁnition of join denote dimension distinct points element of G equivalent etry Euclidean 3-space Euclidean geometry example extension factor system familiar ﬁnd ﬁnite set ﬁrst formal geom group theory half-space holds homomorphism idempotent law identity element implies independent inﬁnite dimensional interior intersection inverse inverse elements join operation join system Lemma linear closure linear space Mathematical multiplication notion open convex set opposite cosets partially ordered ﬁeld plane Projective Geometry Proof rays real numbers relation satisﬁes J1 satisﬁes Postulate satisfy Section segment set of points signiﬁcance Similarly single element subset subsystem system G Theorem 23 Theorem 36 theory of join trichotomy law unique University of Buffalo valid vector veriﬁed xCab yields