THE PURPOSE OF THIS PAPER IS TO COMPUTING OF FUNDAMENTAL RELATIONS OF (GENERAL) HYPERGROUPS.IN THIS REGARDS FIRST WE STUDY SOME BASIC PROPERTIES OF FUNDAMENTAL RELATION OF HYPERGROUPS, THEN WE SHOW THAT ANY GIVEN GROUP IS ISOMORPHISM TO THE FUNDAMENTAL GROUP OF A NONTRIVIAL HYPERGROUP.ESPECIALLY ANY ABELIAN GROUP IS ISOMORPHIC TO A FUNDAMENTAL GROUP OF A COMMUTATIVE HYPERGROUP. FINALLY WE STUDY THE CONNECTIONS BETWEEN CATEGORIES OF HYPERGROUPS AND GROUPS VIA THE FUNDAMENTAL RELATION.

IN THIS PAPER, WE CONSIDER THE NOTIONS OF TM-ALGEBRAS AND QUASI HYPER TCK-ALGEBRAS, GIVE SOME RELATED RESULTS, INTRODUCE THE RELATION B ON HYPER BCK-ALGEBRAS AND LET B* BE THE TRANSITIVE CLOSURE OFB. THEN BY CONSIDERING THE CONCEPT OF STRONGLY REGULAR EQUIVALENCE RELATION (FUNDAMENTAL RELATION) B* ON QUASI HYPER BCK-ALGEBRAS, WE DEFINE THE NOTION OF FUNDAMENTAL TM-ALGEBRA AND WE PROVE THAT ANY COUNTABLE TM-ALGEBRA IS A FUNDAMENTAL TM-ALGEBRA AND INFINITE COUNTABLE TM-ALGEBRA IS A FUNDAMENTAL TM-ALGEBRA OF ITSELF.