SUPPOSE THAT E AND F ARE HILBERT C∗-MODULES OVER C∗-ALGEBRAS A AND B, RESPECTIVELY. LET Φ: A → B BE A MORPHISMOF C∗-ALGEBRAS AND F: E → F BE A Φ-MORPHISM OF HILBERT C∗-MODULES. IN THIS TALK, WE INVESTIGATE UNDER WHICH CONDITIONS THE Φ-MORPHISMS OF HILBERT C∗-MODULES PRESERVE BIRKHOFF ORTHOGONALITY.

THIS ARTICLE, WE START WITH A COLLECTION  M OF MORPHISMS OF A SMALL CATEGORY X, THAT SATISFIES CERTAIN CONDITIONS AND WE CONSTRUCT AN UNIVERSAL CLOSURE OPERATION BY GENERAL METHOD. NEXT DESCRIBE CLOSED SUB OBJECTS RELATIVE TO THIS UNIVERSAL CLOSURE BY A DIFFERENT VIEW POINT.

LET G BE A FINITE P-GROUP, WHERE P IS A PRIME NUMBER AND AUT(G) BE THE AUTOMORPHISM GROUP OF G. WE PROVE THAT IF AUT(G) IS ISOMORPHIC TO GL(N, P) FOR SOME POSITIVE INTEGER N, THEN G IS AN ELEMENTARY ABELIAN GROUP OF ORDER PN. MOREOVER WE PROVE THAT IF G IS A FINITE ABELIAN P-GROUPS WITH |AUT(G)| = |GL(D, P)|, THEN G IS AN ELEMENTARY ABELIAN GROUP OF ORDER PD, WHERE D IS THE MINIMAL NUMBER OF GENERATORS OF G.

ACCORDING TO A THEOREM BY PENROSE AN ACHRONAL BOUNDARY B SPLITS THE SPACE-TIME INTO 3 DISJOINT SETS. B ITSELF, A PAST SET P AND A FUTURE SET F SUCH THAT ¶P=¶F=B. IN MINKOWSKI SPACE-TIME IT ALWAYS TURNS OUT THAT F=I+(B) AND P=I-(B).BUT THIS NEED NOT HOLD FOR ALL SPACES. HERE WE SHOW THAT THE NECESSARY AND SUFFICIENT CONDITION FOR THIS EQUATION TO HOLD IS THAT B BE A MAXIMAL ACHRONAL BOUNDARY.