Some Studies On Thompson-Higman Monoids And Prefix Codes

This main research of article is in the Thompson-Higman monoids and prefix codes of the semigroup algebra theory. A monoid is a semigroup with identity1, monoid is equivalent to a bridge between semigroup and group.Though the research methods of the semigroup derive from the group, but due to differences of their scope. Semigroup has a great deal of difference from research objects to research methods and findings with the group.Semigroup theory showes great superiority in applications.In particular, It is widely used in the field of coding theory, cryptography, sensor etc.So far the semigroup algebra theory has been studied for more than60years, and it gives birth to many of the emerging disciplines, like formal language, cryptography, automata theory, coding theory. In turn, these disciplines promote the development of the semigroup algebra theory.The semigroup structure is a key point of the semigroup algebra theory. One of the researching importance of the semigroup theory is the structure of semigroup. Presently the structure research of semigroup which is abundant is regular semigroup. So far the structure of the regular semigroups obtaine most abundant results by the study. Green’s relations (by J.A.Green1951) is a most critical tool of studying regular semigroup structure,and it plays a fundamental role in the development of the semigroup theory1. The first chapter mainly summarized the research background, research status and future development trends of the semigroup algebra theory,.and gives a statement to some basic knowledge and basic concepts of the semigroup theory, and gives two algebraic theory of semigroups equivalence relations:equivalence relations and congruence relations.At last we give the definitions of the semilattice and strong semilattice.2.The Green relations are generalized in the unsymmetric form.cryptic r-superample semigroups are studied by utlizing the generalized Green relations, we show that cryptic r-superample semigroups are semilattice of completely J*,～-simple semigroups and a cryptic r-superample semigroup is a normal cryptic r-superample semigroup if and only it is a strong semilattice of completely J*,～-simple semigroups. we give a regular r-superabundant semigroup nature of the theorem, this theorem by M.Petrich in completely regular semigroupsfamous theorem.Namely:A super r-ample semigroup S is a regular super r-ample semigroup if and only if it is a G-strong semilattice of completely J*,～-simple semigroups.3. The knowledge of the prefix code and the Thompson-Higman group are introduced in the third chapter. Richard J. Thompson groups are well-known because of their significant properties, Graham, Higman extended to a groups(k≥2, k≥i≥1)of a similar nature, they expressed an infinite simple group limited, which includes all finite group.A lot of references can be investigated on the Thompson-Higman groups. The group can also be extended to the monoids, and the two have many similar properties.This is a new field of study with the prefix codes links closely and therefore it has important research value.The method of researching group and semigroup is different, but if we generalize the nature of the group to the monoid, namely the research method of group and semigroup is connected, and monoid and prefix codes contact close, so studying monoid can find some very important properties of prefix codes, so it is very important.